From 6a08df1fba0b96de0ef2cc5223c1cd7943c3e6d3 Mon Sep 17 00:00:00 2001
From: Morten Hjorth-Jensen <morten.hjorth-jensen@fys.uio.no>
Date: Mon, 5 Sep 2022 21:35:25 +0200
Subject: [PATCH] typos in week 35

---
 doc/pub/week35/html/._week35-bs000.html      |   2 +-
 doc/pub/week35/html/._week35-bs050.html      |  10 +-
 doc/pub/week35/html/week35-bs.html           |   2 +-
 doc/pub/week35/html/week35-reveal.html       |  14 +-
 doc/pub/week35/html/week35-solarized.html    |  12 +-
 doc/pub/week35/html/week35.html              |  12 +-
 doc/pub/week35/ipynb/ipynb-week35-src.tar.gz | Bin 191 -> 192 bytes
 doc/pub/week35/ipynb/week35.ipynb            | 858 +++++++++----------
 doc/src/week35/week35.do.txt                 |  11 +-
 9 files changed, 433 insertions(+), 488 deletions(-)

diff --git a/doc/pub/week35/html/._week35-bs000.html b/doc/pub/week35/html/._week35-bs000.html
index 6c5f4a560..edf5fd391 100644
--- a/doc/pub/week35/html/._week35-bs000.html
+++ b/doc/pub/week35/html/._week35-bs000.html
@@ -404,7 +404,7 @@ MathJax.Hub.Config({
 </center>
 <br>
 <center>
-<h4>Sep 2, 2022</h4>
+<h4>Sep 5, 2022</h4>
 </center> <!-- date -->
 <br>
 
diff --git a/doc/pub/week35/html/._week35-bs050.html b/doc/pub/week35/html/._week35-bs050.html
index e082f1b15..c39729458 100644
--- a/doc/pub/week35/html/._week35-bs050.html
+++ b/doc/pub/week35/html/._week35-bs050.html
@@ -402,12 +402,6 @@ $$
 
 <p>We define \( \boldsymbol{\Sigma}^T\boldsymbol{\Sigma}=\tilde{\boldsymbol{\Sigma}}^2 \) which is  a diagonal matrix containing only the singular values squared. It has dimensionality \( p \times p \).</p>
 
-<p>This means, using the orthogonality of \( \boldsymbol{V} \),  that we get</p>
-
-$$
-\boldsymbol{X}^T\boldsymbol{X}=\tilde{\boldsymbol{\Sigma}}^2.
-$$
-
 <p>We can now insert the result for the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) into our equation for ordinary least squares where</p>
 
 $$
@@ -417,10 +411,10 @@ $$
 <p>and using our SVD decomposition of \( \boldsymbol{X} \) we have</p>
 
 $$
-\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\tilde{\boldsymbol{\Sigma}}^{-2}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
+\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\left(\boldsymbol{V}\tilde{\boldsymbol{\Sigma}}^{2}(\boldsymbol{V}^T\right)^{-1}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
 $$
 
-<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),</p>
+<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),,</p>
 
 $$
 \tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_j\boldsymbol{y},
diff --git a/doc/pub/week35/html/week35-bs.html b/doc/pub/week35/html/week35-bs.html
index 6c5f4a560..edf5fd391 100644
--- a/doc/pub/week35/html/week35-bs.html
+++ b/doc/pub/week35/html/week35-bs.html
@@ -404,7 +404,7 @@ MathJax.Hub.Config({
 </center>
 <br>
 <center>
-<h4>Sep 2, 2022</h4>
+<h4>Sep 5, 2022</h4>
 </center> <!-- date -->
 <br>
 
diff --git a/doc/pub/week35/html/week35-reveal.html b/doc/pub/week35/html/week35-reveal.html
index cff9aa38e..a879fdbeb 100644
--- a/doc/pub/week35/html/week35-reveal.html
+++ b/doc/pub/week35/html/week35-reveal.html
@@ -184,7 +184,7 @@ MathJax.Hub.Config({
 </center>
 <br>
 <center>
-<h4>Sep 2, 2022</h4>
+<h4>Sep 5, 2022</h4>
 </center> <!-- date -->
 <br>
 
@@ -2538,14 +2538,6 @@ $$
 
 <p>We define \( \boldsymbol{\Sigma}^T\boldsymbol{\Sigma}=\tilde{\boldsymbol{\Sigma}}^2 \) which is  a diagonal matrix containing only the singular values squared. It has dimensionality \( p \times p \).</p>
 
-<p>This means, using the orthogonality of \( \boldsymbol{V} \),  that we get</p>
-
-<p>&nbsp;<br>
-$$
-\boldsymbol{X}^T\boldsymbol{X}=\tilde{\boldsymbol{\Sigma}}^2.
-$$
-<p>&nbsp;<br>
-
 <p>We can now insert the result for the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) into our equation for ordinary least squares where</p>
 
 <p>&nbsp;<br>
@@ -2558,11 +2550,11 @@ $$
 
 <p>&nbsp;<br>
 $$
-\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\tilde{\boldsymbol{\Sigma}}^{-2}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
+\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\left(\boldsymbol{V}\tilde{\boldsymbol{\Sigma}}^{2}(\boldsymbol{V}^T\right)^{-1}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
 $$
 <p>&nbsp;<br>
 
-<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),</p>
+<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),,</p>
 
 <p>&nbsp;<br>
 $$
diff --git a/doc/pub/week35/html/week35-solarized.html b/doc/pub/week35/html/week35-solarized.html
index d493bcd33..9b0c076b4 100644
--- a/doc/pub/week35/html/week35-solarized.html
+++ b/doc/pub/week35/html/week35-solarized.html
@@ -330,7 +330,7 @@ MathJax.Hub.Config({
 </center>
 <br>
 <center>
-<h4>Sep 2, 2022</h4>
+<h4>Sep 5, 2022</h4>
 </center> <!-- date -->
 <br>
 
@@ -2513,12 +2513,6 @@ $$
 
 <p>We define \( \boldsymbol{\Sigma}^T\boldsymbol{\Sigma}=\tilde{\boldsymbol{\Sigma}}^2 \) which is  a diagonal matrix containing only the singular values squared. It has dimensionality \( p \times p \).</p>
 
-<p>This means, using the orthogonality of \( \boldsymbol{V} \),  that we get</p>
-
-$$
-\boldsymbol{X}^T\boldsymbol{X}=\tilde{\boldsymbol{\Sigma}}^2.
-$$
-
 <p>We can now insert the result for the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) into our equation for ordinary least squares where</p>
 
 $$
@@ -2528,10 +2522,10 @@ $$
 <p>and using our SVD decomposition of \( \boldsymbol{X} \) we have</p>
 
 $$
-\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\tilde{\boldsymbol{\Sigma}}^{-2}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
+\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\left(\boldsymbol{V}\tilde{\boldsymbol{\Sigma}}^{2}(\boldsymbol{V}^T\right)^{-1}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
 $$
 
-<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),</p>
+<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),,</p>
 
 $$
 \tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_j\boldsymbol{y},
diff --git a/doc/pub/week35/html/week35.html b/doc/pub/week35/html/week35.html
index f3d317134..36d50bc9a 100644
--- a/doc/pub/week35/html/week35.html
+++ b/doc/pub/week35/html/week35.html
@@ -407,7 +407,7 @@ MathJax.Hub.Config({
 </center>
 <br>
 <center>
-<h4>Sep 2, 2022</h4>
+<h4>Sep 5, 2022</h4>
 </center> <!-- date -->
 <br>
 
@@ -2590,12 +2590,6 @@ $$
 
 <p>We define \( \boldsymbol{\Sigma}^T\boldsymbol{\Sigma}=\tilde{\boldsymbol{\Sigma}}^2 \) which is  a diagonal matrix containing only the singular values squared. It has dimensionality \( p \times p \).</p>
 
-<p>This means, using the orthogonality of \( \boldsymbol{V} \),  that we get</p>
-
-$$
-\boldsymbol{X}^T\boldsymbol{X}=\tilde{\boldsymbol{\Sigma}}^2.
-$$
-
 <p>We can now insert the result for the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) into our equation for ordinary least squares where</p>
 
 $$
@@ -2605,10 +2599,10 @@ $$
 <p>and using our SVD decomposition of \( \boldsymbol{X} \) we have</p>
 
 $$
-\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\tilde{\boldsymbol{\Sigma}}^{-2}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
+\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\left(\boldsymbol{V}\tilde{\boldsymbol{\Sigma}}^{2}(\boldsymbol{V}^T\right)^{-1}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
 $$
 
-<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),</p>
+<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),,</p>
 
 $$
 \tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_j\boldsymbol{y},
diff --git a/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz b/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz
index 67da7d35dbb3d9ca46cd7bd1701c51dc64ed4a05..c88901ef6c4591bdbc96696d2b22e91a22562a9b 100644
GIT binary patch
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DIo~~x

diff --git a/doc/pub/week35/ipynb/week35.ipynb b/doc/pub/week35/ipynb/week35.ipynb
index 97afac994..94dc68e0b 100644
--- a/doc/pub/week35/ipynb/week35.ipynb
+++ b/doc/pub/week35/ipynb/week35.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "6d0d3ab1",
+   "id": "7a504e7b",
    "metadata": {
     "editable": true
    },
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d2a5af6f",
+   "id": "a7356dcf",
    "metadata": {
     "editable": true
    },
@@ -22,14 +22,14 @@
     "# Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression\n",
     "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
     "\n",
-    "Date: **Sep 2, 2022**\n",
+    "Date: **Sep 5, 2022**\n",
     "\n",
     "Copyright 1999-2022, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "4b1d863c",
+   "id": "009a2cd7",
    "metadata": {
     "editable": true
    },
@@ -49,7 +49,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "eb99cb95",
+   "id": "fa128d21",
    "metadata": {
     "editable": true
    },
@@ -69,7 +69,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a6bb0d0d",
+   "id": "ec55c75b",
    "metadata": {
     "editable": true
    },
@@ -90,7 +90,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0471e8e7",
+   "id": "8669f9d4",
    "metadata": {
     "editable": true
    },
@@ -124,7 +124,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0363011a",
+   "id": "d7601ed3",
    "metadata": {
     "editable": true
    },
@@ -146,7 +146,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4aa5c1f9",
+   "id": "6c70e674",
    "metadata": {
     "editable": true
    },
@@ -174,7 +174,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3e39c550",
+   "id": "3abfcfe1",
    "metadata": {
     "editable": true
    },
@@ -189,7 +189,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4d67106b",
+   "id": "502435c8",
    "metadata": {
     "editable": true
    },
@@ -201,7 +201,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0af16b7b",
+   "id": "2169e3af",
    "metadata": {
     "editable": true
    },
@@ -216,7 +216,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "23a712fa",
+   "id": "b1551b35",
    "metadata": {
     "editable": true
    },
@@ -229,7 +229,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "61d333ef",
+   "id": "114fd017",
    "metadata": {
     "editable": true
    },
@@ -241,7 +241,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8c3da8ec",
+   "id": "a60d07ea",
    "metadata": {
     "editable": true
    },
@@ -251,7 +251,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d797b999",
+   "id": "e8645cab",
    "metadata": {
     "editable": true
    },
@@ -262,7 +262,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9c6a2274",
+   "id": "932554ac",
    "metadata": {
     "editable": true
    },
@@ -280,7 +280,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "af5a7b11",
+   "id": "fb9cea35",
    "metadata": {
     "editable": true
    },
@@ -291,7 +291,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "478d5dd9",
+   "id": "06b38aad",
    "metadata": {
     "editable": true
    },
@@ -303,7 +303,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "db469d4e",
+   "id": "1e26ff80",
    "metadata": {
     "editable": true
    },
@@ -313,7 +313,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8ef99c65",
+   "id": "14f74024",
    "metadata": {
     "editable": true
    },
@@ -325,7 +325,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3ac9c6bb",
+   "id": "3957edbf",
    "metadata": {
     "editable": true
    },
@@ -335,7 +335,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "76dfd1de",
+   "id": "9c32e2a4",
    "metadata": {
     "editable": true
    },
@@ -347,7 +347,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "86c7c903",
+   "id": "0c3a77b8",
    "metadata": {
     "editable": true
    },
@@ -357,7 +357,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3c45ca30",
+   "id": "1b4aa107",
    "metadata": {
     "editable": true
    },
@@ -376,7 +376,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f73d4ebd",
+   "id": "e3054f44",
    "metadata": {
     "editable": true
    },
@@ -386,7 +386,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ff68d0c2",
+   "id": "38978a8d",
    "metadata": {
     "editable": true
    },
@@ -398,7 +398,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b8858121",
+   "id": "599bbd4d",
    "metadata": {
     "editable": true
    },
@@ -408,7 +408,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e9087a01",
+   "id": "e78f5e77",
    "metadata": {
     "editable": true
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@@ -424,7 +424,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a2a04531",
+   "id": "2165c2d9",
    "metadata": {
     "editable": true
    },
@@ -444,7 +444,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2c9b9f9a",
+   "id": "97d08900",
    "metadata": {
     "editable": true
    },
@@ -454,7 +454,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "080d030c",
+   "id": "0aafeeac",
    "metadata": {
     "editable": true
    },
@@ -465,7 +465,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2da15ad1",
+   "id": "7f230e6b",
    "metadata": {
     "editable": true
    },
@@ -484,7 +484,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "529600dd",
+   "id": "de1c4d31",
    "metadata": {
     "editable": true
    },
@@ -494,7 +494,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f78ba3a1",
+   "id": "35078725",
    "metadata": {
     "editable": true
    },
@@ -506,7 +506,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cebda954",
+   "id": "923dcb9d",
    "metadata": {
     "editable": true
    },
@@ -516,7 +516,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "591b9b05",
+   "id": "5bbd6523",
    "metadata": {
     "editable": true
    },
@@ -527,7 +527,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f2b22899",
+   "id": "7c434251",
    "metadata": {
     "editable": true
    },
@@ -547,7 +547,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "96033227",
+   "id": "54f3dbb1",
    "metadata": {
     "editable": true
    },
@@ -559,7 +559,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a3d801ef",
+   "id": "bc649187",
    "metadata": {
     "editable": true
    },
@@ -574,7 +574,7 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "339c754e",
+   "id": "c7d141f9",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -656,7 +656,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dc1ed6c0",
+   "id": "e20e3a48",
    "metadata": {
     "editable": true
    },
@@ -666,7 +666,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b2a748d5",
+   "id": "cdffdc10",
    "metadata": {
     "editable": true
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@@ -678,7 +678,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "51b1434d",
+   "id": "06de0218",
    "metadata": {
     "editable": true
    },
@@ -688,7 +688,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f80990bf",
+   "id": "cfa564be",
    "metadata": {
     "editable": true
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@@ -699,7 +699,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c20481c8",
+   "id": "02116d1e",
    "metadata": {
     "editable": true
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@@ -711,7 +711,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9f1c3820",
+   "id": "9e33c20f",
    "metadata": {
     "editable": true
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@@ -721,7 +721,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5967d99",
+   "id": "cc6a84c9",
    "metadata": {
     "editable": true
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@@ -733,7 +733,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "63d17ff2",
+   "id": "487975c0",
    "metadata": {
     "editable": true
    },
@@ -743,7 +743,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "12dbd290",
+   "id": "1a91e724",
    "metadata": {
     "editable": true
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@@ -755,7 +755,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1504085a",
+   "id": "d294daff",
    "metadata": {
     "editable": true
    },
@@ -768,7 +768,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "40182aa6",
+   "id": "f183fdd0",
    "metadata": {
     "editable": true
    },
@@ -780,7 +780,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f9de3192",
+   "id": "b0210430",
    "metadata": {
     "editable": true
    },
@@ -790,7 +790,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "69931339",
+   "id": "6d0be46e",
    "metadata": {
     "editable": true
    },
@@ -802,7 +802,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f248f666",
+   "id": "362ea346",
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@@ -814,7 +814,7 @@
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@@ -825,7 +825,7 @@
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@@ -837,7 +837,7 @@
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@@ -856,7 +856,7 @@
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@@ -869,7 +869,7 @@
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@@ -879,7 +879,7 @@
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@@ -891,7 +891,7 @@
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@@ -901,7 +901,7 @@
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@@ -913,7 +913,7 @@
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@@ -923,7 +923,7 @@
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@@ -935,7 +935,7 @@
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@@ -946,7 +946,7 @@
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@@ -958,7 +958,7 @@
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@@ -968,7 +968,7 @@
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@@ -980,7 +980,7 @@
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@@ -990,7 +990,7 @@
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@@ -1002,7 +1002,7 @@
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@@ -1023,7 +1023,7 @@
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@@ -1036,7 +1036,7 @@
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@@ -1048,7 +1048,7 @@
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@@ -1060,7 +1060,7 @@
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@@ -1072,7 +1072,7 @@
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@@ -1084,7 +1084,7 @@
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@@ -1094,7 +1094,7 @@
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@@ -1110,7 +1110,7 @@
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@@ -1122,7 +1122,7 @@
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@@ -1132,7 +1132,7 @@
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@@ -1144,7 +1144,7 @@
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@@ -1162,7 +1162,7 @@
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@@ -1173,7 +1173,7 @@
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@@ -1185,7 +1185,7 @@
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@@ -1195,7 +1195,7 @@
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@@ -1207,7 +1207,7 @@
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@@ -1217,7 +1217,7 @@
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@@ -1229,7 +1229,7 @@
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@@ -1239,7 +1239,7 @@
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@@ -1253,7 +1253,7 @@
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@@ -1268,7 +1268,7 @@
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@@ -1279,7 +1279,7 @@
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@@ -1292,7 +1292,7 @@
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@@ -1303,7 +1303,7 @@
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@@ -1326,7 +1326,7 @@
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@@ -1340,7 +1340,7 @@
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@@ -1353,7 +1353,7 @@
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@@ -1364,7 +1364,7 @@
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@@ -1376,7 +1376,7 @@
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@@ -1387,7 +1387,7 @@
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@@ -1403,7 +1403,7 @@
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@@ -1414,7 +1414,7 @@
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@@ -1428,7 +1428,7 @@
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@@ -1449,7 +1449,7 @@
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@@ -1460,7 +1460,7 @@
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@@ -1509,7 +1509,7 @@
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@@ -1520,7 +1520,7 @@
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@@ -1545,7 +1545,7 @@
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@@ -1557,7 +1557,7 @@
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@@ -1601,7 +1601,7 @@
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@@ -1613,7 +1613,7 @@
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@@ -1629,7 +1629,7 @@
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@@ -1640,7 +1640,7 @@
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@@ -1658,7 +1658,7 @@
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@@ -1669,7 +1669,7 @@
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@@ -1683,7 +1683,7 @@
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@@ -1694,7 +1694,7 @@
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@@ -1707,7 +1707,7 @@
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@@ -1718,7 +1718,7 @@
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@@ -1735,7 +1735,7 @@
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@@ -1746,7 +1746,7 @@
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@@ -1762,7 +1762,7 @@
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@@ -1773,7 +1773,7 @@
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@@ -1797,7 +1797,7 @@
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@@ -1808,7 +1808,7 @@
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@@ -1821,7 +1821,7 @@
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@@ -1832,7 +1832,7 @@
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@@ -1852,7 +1852,7 @@
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@@ -1863,7 +1863,7 @@
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@@ -1906,7 +1906,7 @@
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@@ -1921,7 +1921,7 @@
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@@ -1950,7 +1950,7 @@
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@@ -1975,7 +1975,7 @@
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@@ -1995,7 +1995,7 @@
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@@ -2022,7 +2022,7 @@
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@@ -2035,7 +2035,7 @@
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@@ -2047,7 +2047,7 @@
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@@ -2058,7 +2058,7 @@
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@@ -2074,7 +2074,7 @@
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@@ -2108,7 +2108,7 @@
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@@ -2118,7 +2118,7 @@
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@@ -2133,7 +2133,7 @@
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@@ -2145,7 +2145,7 @@
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@@ -2155,7 +2155,7 @@
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@@ -2171,7 +2171,7 @@
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@@ -2188,7 +2188,7 @@
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@@ -2201,7 +2201,7 @@
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@@ -2248,7 +2248,7 @@
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@@ -2259,7 +2259,7 @@
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@@ -2362,7 +2362,7 @@
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@@ -2376,7 +2376,7 @@
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@@ -2388,7 +2388,7 @@
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-   "id": "7641c761",
+   "id": "140e6cda",
    "metadata": {
     "editable": true
    },
@@ -2400,7 +2400,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "98ce47f4",
+   "id": "d1ced789",
    "metadata": {
     "editable": true
    },
@@ -2412,7 +2412,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b135ee8e",
+   "id": "1921cc4b",
    "metadata": {
     "editable": true
    },
@@ -2422,7 +2422,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d410a00b",
+   "id": "b9deae5b",
    "metadata": {
     "editable": true
    },
@@ -2434,7 +2434,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "10319a09",
+   "id": "542878b0",
    "metadata": {
     "editable": true
    },
@@ -2444,7 +2444,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f29c83d4",
+   "id": "b9937385",
    "metadata": {
     "editable": true
    },
@@ -2456,7 +2456,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4d2dcc6c",
+   "id": "7d6d879e",
    "metadata": {
     "editable": true
    },
@@ -2467,7 +2467,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5eb4eeb6",
+   "id": "f9d858c3",
    "metadata": {
     "editable": true
    },
@@ -2479,7 +2479,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8981743f",
+   "id": "54f3ca73",
    "metadata": {
     "editable": true
    },
@@ -2491,7 +2491,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e0be6912",
+   "id": "fca76dcf",
    "metadata": {
     "editable": true
    },
@@ -2501,7 +2501,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b5c400a2",
+   "id": "e5528b9b",
    "metadata": {
     "editable": true
    },
@@ -2513,7 +2513,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dfd10d49",
+   "id": "b02ae145",
    "metadata": {
     "editable": true
    },
@@ -2525,7 +2525,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f2c52066",
+   "id": "11f70a5e",
    "metadata": {
     "editable": true
    },
@@ -2535,7 +2535,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7b503fe6",
+   "id": "0bc7a63e",
    "metadata": {
     "editable": true
    },
@@ -2547,7 +2547,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5d2f1d89",
+   "id": "4526b0a9",
    "metadata": {
     "editable": true
    },
@@ -2557,7 +2557,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7affe5a3",
+   "id": "86c48422",
    "metadata": {
     "editable": true
    },
@@ -2569,7 +2569,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2e91c7a6",
+   "id": "d064d03b",
    "metadata": {
     "editable": true
    },
@@ -2579,7 +2579,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3422455f",
+   "id": "cf173e38",
    "metadata": {
     "editable": true
    },
@@ -2619,7 +2619,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b137cd41",
+   "id": "ca0d88c0",
    "metadata": {
     "editable": true
    },
@@ -2636,7 +2636,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "73300817",
+   "id": "d7f5eb25",
    "metadata": {
     "editable": true
    },
@@ -2659,7 +2659,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d25633a5",
+   "id": "f4102dad",
    "metadata": {
     "editable": true
    },
@@ -2676,7 +2676,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ec6c8cef",
+   "id": "2bfa2b39",
    "metadata": {
     "editable": true
    },
@@ -2695,7 +2695,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a30834c8",
+   "id": "fae2ede0",
    "metadata": {
     "editable": true
    },
@@ -2706,7 +2706,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3e7cc8a5",
+   "id": "61b75442",
    "metadata": {
     "editable": true
    },
@@ -2718,7 +2718,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3f1b2e29",
+   "id": "3ab4d4c8",
    "metadata": {
     "editable": true
    },
@@ -2736,7 +2736,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5500017e",
+   "id": "cedfd8e0",
    "metadata": {
     "editable": true
    },
@@ -2752,7 +2752,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1445c29c",
+   "id": "562d7a68",
    "metadata": {
     "editable": true
    },
@@ -2764,7 +2764,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "42071b7d",
+   "id": "434a3180",
    "metadata": {
     "editable": true
    },
@@ -2774,7 +2774,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3b65710f",
+   "id": "f1043793",
    "metadata": {
     "editable": true
    },
@@ -2789,7 +2789,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "697f5458",
+   "id": "d2037f30",
    "metadata": {
     "editable": true
    },
@@ -2801,7 +2801,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "63852926",
+   "id": "cd5daa55",
    "metadata": {
     "editable": true
    },
@@ -2811,7 +2811,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b195fc62",
+   "id": "1149a315",
    "metadata": {
     "editable": true
    },
@@ -2823,7 +2823,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "94e886de",
+   "id": "4b13b34c",
    "metadata": {
     "editable": true
    },
@@ -2833,7 +2833,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9ce55ee9",
+   "id": "fd615af3",
    "metadata": {
     "editable": true
    },
@@ -2845,7 +2845,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dbc19a38",
+   "id": "8805ce4f",
    "metadata": {
     "editable": true
    },
@@ -2857,7 +2857,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f1dd6b14",
+   "id": "a26a7410",
    "metadata": {
     "editable": true
    },
@@ -2872,7 +2872,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ea594394",
+   "id": "7f1977e2",
    "metadata": {
     "editable": true
    },
@@ -2883,7 +2883,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9362b6ff",
+   "id": "9b82612a",
    "metadata": {
     "editable": true
    },
@@ -2903,7 +2903,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "412bb04d",
+   "id": "1524a2b0",
    "metadata": {
     "editable": true
    },
@@ -2915,7 +2915,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c4f06e0e",
+   "id": "f0ad147f",
    "metadata": {
     "editable": true
    },
@@ -2925,7 +2925,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4a69f030",
+   "id": "427004ed",
    "metadata": {
     "editable": true
    },
@@ -2937,7 +2937,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a836ce65",
+   "id": "4d1ae4b9",
    "metadata": {
     "editable": true
    },
@@ -2966,7 +2966,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "172c1829",
+   "id": "0696b7c1",
    "metadata": {
     "editable": true
    },
@@ -2993,7 +2993,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8cc89973",
+   "id": "dcc804e5",
    "metadata": {
     "editable": true
    },
@@ -3004,7 +3004,7 @@
   {
    "cell_type": "code",
    "execution_count": 26,
-   "id": "89bce983",
+   "id": "90621d7a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3044,7 +3044,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8eab2044",
+   "id": "95a2af33",
    "metadata": {
     "editable": true
    },
@@ -3061,7 +3061,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "73c9168e",
+   "id": "d2277111",
    "metadata": {
     "editable": true
    },
@@ -3084,7 +3084,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "155f9f57",
+   "id": "2367d702",
    "metadata": {
     "editable": true
    },
@@ -3094,7 +3094,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "82bb3059",
+   "id": "22849bfa",
    "metadata": {
     "editable": true
    },
@@ -3108,7 +3108,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "30514f5e",
+   "id": "eed9fecc",
    "metadata": {
     "editable": true
    },
@@ -3127,7 +3127,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5e9c64c",
+   "id": "42f93c0d",
    "metadata": {
     "editable": true
    },
@@ -3137,7 +3137,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "87d3e730",
+   "id": "2187d642",
    "metadata": {
     "editable": true
    },
@@ -3149,7 +3149,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "86bf0464",
+   "id": "57c4d306",
    "metadata": {
     "editable": true
    },
@@ -3163,7 +3163,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "616d56a5",
+   "id": "51e30a8c",
    "metadata": {
     "editable": true
    },
@@ -3175,7 +3175,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a3b6e39f",
+   "id": "d805988d",
    "metadata": {
     "editable": true
    },
@@ -3185,7 +3185,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3cf2c4bb",
+   "id": "328b2cd8",
    "metadata": {
     "editable": true
    },
@@ -3197,7 +3197,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e1c0ac12",
+   "id": "7562a722",
    "metadata": {
     "editable": true
    },
@@ -3214,7 +3214,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "aa9dec0f",
+   "id": "296222cb",
    "metadata": {
     "editable": true
    },
@@ -3224,7 +3224,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "22f3448a",
+   "id": "f8c01215",
    "metadata": {
     "editable": true
    },
@@ -3240,7 +3240,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e4798ffc",
+   "id": "83c84e38",
    "metadata": {
     "editable": true
    },
@@ -3250,7 +3250,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b08086d2",
+   "id": "6cea090d",
    "metadata": {
     "editable": true
    },
@@ -3266,7 +3266,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "550ba67e",
+   "id": "69feadb2",
    "metadata": {
     "editable": true
    },
@@ -3276,7 +3276,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5c03b1ba",
+   "id": "e2a62928",
    "metadata": {
     "editable": true
    },
@@ -3292,7 +3292,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "61de6fe9",
+   "id": "8902dc97",
    "metadata": {
     "editable": true
    },
@@ -3302,7 +3302,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "38a02093",
+   "id": "c274a1d5",
    "metadata": {
     "editable": true
    },
@@ -3319,7 +3319,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a13230b4",
+   "id": "2cfcf5f8",
    "metadata": {
     "editable": true
    },
@@ -3331,7 +3331,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "42186b3e",
+   "id": "2f00cd88",
    "metadata": {
     "editable": true
    },
@@ -3343,7 +3343,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e3600aa7",
+   "id": "46b73536",
    "metadata": {
     "editable": true
    },
@@ -3355,7 +3355,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "35373cc7",
+   "id": "8ccbf9eb",
    "metadata": {
     "editable": true
    },
@@ -3365,7 +3365,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3af04b6",
+   "id": "3bf8fdb3",
    "metadata": {
     "editable": true
    },
@@ -3377,41 +3377,19 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1f10b10a",
+   "id": "a94157e4",
    "metadata": {
     "editable": true
    },
    "source": [
     "We define $\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma}=\\tilde{\\boldsymbol{\\Sigma}}^2$ which is  a diagonal matrix containing only the singular values squared. It has dimensionality $p \\times p$.\n",
     "\n",
-    "This means, using the orthogonality of $\\boldsymbol{V}$,  that we get"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "68aefa4e",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\boldsymbol{X}^T\\boldsymbol{X}=\\tilde{\\boldsymbol{\\Sigma}}^2.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "0cde90bc",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
     "We can now insert the result for the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$ into our equation for ordinary least squares where"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "cdea96a7",
+   "id": "67301cec",
    "metadata": {
     "editable": true
    },
@@ -3423,7 +3401,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1862ddc3",
+   "id": "4a7d00ed",
    "metadata": {
     "editable": true
    },
@@ -3433,29 +3411,29 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1ca5ba4f",
+   "id": "705eb661",
    "metadata": {
     "editable": true
    },
    "source": [
     "$$\n",
-    "\\tilde{y}_{\\mathrm{OLS}}=\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T\\tilde{\\boldsymbol{\\Sigma}}^{-2}\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{U}^T\\boldsymbol{y},\n",
+    "\\tilde{y}_{\\mathrm{OLS}}=\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T\\left(\\boldsymbol{V}\\tilde{\\boldsymbol{\\Sigma}}^{2}(\\boldsymbol{V}^T\\right)^{-1}\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{U}^T\\boldsymbol{y},\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "980b7aa5",
+   "id": "69c56644",
    "metadata": {
     "editable": true
    },
    "source": [
-    "which gives us, using the orthogonality of the matrices $\\boldsymbol{U}$ and $\\boldsymbol{V}$,"
+    "which gives us, using the orthogonality of the matrices $\\boldsymbol{U}$ and $\\boldsymbol{V}$,,"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "308c6c39",
+   "id": "9ad7c575",
    "metadata": {
     "editable": true
    },
@@ -3467,7 +3445,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ecad476d",
+   "id": "f1842c58",
    "metadata": {
     "editable": true
    },
@@ -3477,7 +3455,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "68157e26",
+   "id": "c0316a8b",
    "metadata": {
     "editable": true
    },
@@ -3489,7 +3467,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0a7928ff",
+   "id": "ca04dc61",
    "metadata": {
     "editable": true
    },
@@ -3501,7 +3479,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "87cb10f3",
+   "id": "4534b33e",
    "metadata": {
     "editable": true
    },
@@ -3513,7 +3491,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1b487e45",
+   "id": "0d9b0878",
    "metadata": {
     "editable": true
    },
@@ -3525,7 +3503,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fe4c7375",
+   "id": "31121c9e",
    "metadata": {
     "editable": true
    },
@@ -3535,7 +3513,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2af9dcff",
+   "id": "de377680",
    "metadata": {
     "editable": true
    },
@@ -3547,7 +3525,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f9bf66fb",
+   "id": "7ad4c901",
    "metadata": {
     "editable": true
    },
@@ -3558,7 +3536,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a2e67382",
+   "id": "b3f13adc",
    "metadata": {
     "editable": true
    },
@@ -3570,7 +3548,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7c44d794",
+   "id": "76eeac30",
    "metadata": {
     "editable": true
    },
@@ -3580,7 +3558,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7a5dc906",
+   "id": "93e23f88",
    "metadata": {
     "editable": true
    },
@@ -3592,7 +3570,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "be1d6c71",
+   "id": "5463eae9",
    "metadata": {
     "editable": true
    },
@@ -3602,7 +3580,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "881cf391",
+   "id": "937acf1b",
    "metadata": {
     "editable": true
    },
@@ -3614,7 +3592,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ebc57307",
+   "id": "ec3c665a",
    "metadata": {
     "editable": true
    },
@@ -3625,7 +3603,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "de44e7be",
+   "id": "3c3c64fb",
    "metadata": {
     "editable": true
    },
@@ -3637,7 +3615,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "adf01fa0",
+   "id": "baf12e06",
    "metadata": {
     "editable": true
    },
@@ -3655,7 +3633,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5e13c634",
+   "id": "83d68c10",
    "metadata": {
     "editable": true
    },
@@ -3671,7 +3649,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3f456c97",
+   "id": "b197919f",
    "metadata": {
     "editable": true
    },
@@ -3683,7 +3661,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7a5d5cb4",
+   "id": "4d52fd8d",
    "metadata": {
     "editable": true
    },
@@ -3695,7 +3673,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d1dc0e55",
+   "id": "873fe256",
    "metadata": {
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@@ -3707,7 +3685,7 @@
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@@ -3720,7 +3698,7 @@
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@@ -3736,7 +3714,7 @@
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@@ -3750,7 +3728,7 @@
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@@ -3760,7 +3738,7 @@
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@@ -3772,7 +3750,7 @@
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@@ -3782,7 +3760,7 @@
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@@ -3794,7 +3772,7 @@
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@@ -3804,7 +3782,7 @@
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@@ -3818,7 +3796,7 @@
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@@ -3835,7 +3813,7 @@
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@@ -3851,7 +3829,7 @@
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@@ -3863,7 +3841,7 @@
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@@ -3876,7 +3854,7 @@
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@@ -3890,7 +3868,7 @@
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@@ -3900,7 +3878,7 @@
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@@ -3913,7 +3891,7 @@
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@@ -3932,7 +3910,7 @@
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@@ -3944,7 +3922,7 @@
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@@ -3956,7 +3934,7 @@
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@@ -3966,7 +3944,7 @@
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@@ -3978,7 +3956,7 @@
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@@ -3991,7 +3969,7 @@
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@@ -4010,7 +3988,7 @@
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@@ -4020,7 +3998,7 @@
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@@ -4039,7 +4017,7 @@
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@@ -4057,7 +4035,7 @@
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@@ -4071,7 +4049,7 @@
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@@ -4086,7 +4064,7 @@
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@@ -4107,7 +4085,7 @@
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@@ -4124,7 +4102,7 @@
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@@ -4156,7 +4134,7 @@
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@@ -4170,7 +4148,7 @@
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@@ -4183,7 +4161,7 @@
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@@ -4208,7 +4186,7 @@
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@@ -4218,7 +4196,7 @@
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@@ -4229,7 +4207,7 @@
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@@ -4283,7 +4261,7 @@
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@@ -4300,7 +4278,7 @@
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@@ -4312,7 +4290,7 @@
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@@ -4324,7 +4302,7 @@
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@@ -4334,7 +4312,7 @@
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@@ -4351,7 +4329,7 @@
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@@ -4361,7 +4339,7 @@
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@@ -4376,7 +4354,7 @@
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@@ -4386,7 +4364,7 @@
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@@ -4400,7 +4378,7 @@
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@@ -4412,7 +4390,7 @@
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@@ -4424,7 +4402,7 @@
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@@ -4436,7 +4414,7 @@
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@@ -4446,7 +4424,7 @@
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@@ -4458,7 +4436,7 @@
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@@ -4468,7 +4446,7 @@
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@@ -4485,7 +4463,7 @@
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@@ -4495,7 +4473,7 @@
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@@ -4507,7 +4485,7 @@
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@@ -4517,7 +4495,7 @@
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@@ -4529,7 +4507,7 @@
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@@ -4543,7 +4521,7 @@
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@@ -4555,7 +4533,7 @@
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@@ -4577,7 +4555,7 @@
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@@ -4589,7 +4567,7 @@
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@@ -4604,7 +4582,7 @@
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@@ -4616,7 +4594,7 @@
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@@ -4628,7 +4606,7 @@
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@@ -4638,7 +4616,7 @@
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@@ -4650,7 +4628,7 @@
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@@ -4660,7 +4638,7 @@
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@@ -4672,7 +4650,7 @@
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@@ -4682,7 +4660,7 @@
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@@ -4694,7 +4672,7 @@
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@@ -4711,7 +4689,7 @@
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@@ -4724,7 +4702,7 @@
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@@ -4736,7 +4714,7 @@
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@@ -4746,7 +4724,7 @@
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@@ -4759,7 +4737,7 @@
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@@ -4769,7 +4747,7 @@
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@@ -4781,7 +4759,7 @@
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@@ -4794,7 +4772,7 @@
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@@ -4807,7 +4785,7 @@
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@@ -4819,7 +4797,7 @@
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@@ -4831,7 +4809,7 @@
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@@ -4841,7 +4819,7 @@
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@@ -4854,7 +4832,7 @@
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@@ -4866,7 +4844,7 @@
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@@ -4878,7 +4856,7 @@
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@@ -4890,7 +4868,7 @@
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@@ -4902,7 +4880,7 @@
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@@ -4916,7 +4894,7 @@
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@@ -4928,7 +4906,7 @@
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@@ -4938,7 +4916,7 @@
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@@ -4950,7 +4928,7 @@
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@@ -4962,7 +4940,7 @@
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@@ -4974,7 +4952,7 @@
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@@ -4986,7 +4964,7 @@
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@@ -4998,7 +4976,7 @@
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@@ -5017,7 +4995,7 @@
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@@ -5029,7 +5007,7 @@
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@@ -5039,7 +5017,7 @@
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@@ -5051,7 +5029,7 @@
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@@ -5061,7 +5039,7 @@
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@@ -5073,7 +5051,7 @@
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-   "id": "a28fc876",
+   "id": "a90718da",
    "metadata": {
     "editable": true
    },
@@ -5085,7 +5063,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ca7ea691",
+   "id": "f338bae2",
    "metadata": {
     "editable": true
    },
@@ -5101,7 +5079,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ae7ab481",
+   "id": "742b7067",
    "metadata": {
     "editable": true
    },
@@ -5113,7 +5091,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "97f9af9b",
+   "id": "8f35782c",
    "metadata": {
     "editable": true
    },
@@ -5125,7 +5103,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4b462a34",
+   "id": "4847e18c",
    "metadata": {
     "editable": true
    },
@@ -5135,7 +5113,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "18d72942",
+   "id": "bd95701b",
    "metadata": {
     "editable": true
    },
@@ -5147,7 +5125,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "da50edc8",
+   "id": "6e1f03f7",
    "metadata": {
     "editable": true
    },
@@ -5157,7 +5135,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6a05e915",
+   "id": "a802e415",
    "metadata": {
     "editable": true
    },
@@ -5169,7 +5147,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "965ecfb7",
+   "id": "a01a2fc7",
    "metadata": {
     "editable": true
    },
@@ -5186,7 +5164,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0d01ed18",
+   "id": "c6777a4d",
    "metadata": {
     "editable": true
    },
@@ -5198,7 +5176,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c709c1e2",
+   "id": "0b9a3ae5",
    "metadata": {
     "editable": true
    },
@@ -5210,7 +5188,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a0c04297",
+   "id": "bd856f81",
    "metadata": {
     "editable": true
    },
@@ -5220,7 +5198,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1737a954",
+   "id": "9f2080fe",
    "metadata": {
     "editable": true
    },
@@ -5232,7 +5210,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "be59baa0",
+   "id": "cf7ef832",
    "metadata": {
     "editable": true
    },
@@ -5242,7 +5220,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "abefb98a",
+   "id": "fe85bb55",
    "metadata": {
     "editable": true
    },
@@ -5254,7 +5232,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b02f5528",
+   "id": "85cfe1ee",
    "metadata": {
     "editable": true
    },
@@ -5264,7 +5242,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3aa84208",
+   "id": "38f8d8a7",
    "metadata": {
     "editable": true
    },
@@ -5276,7 +5254,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3bcb56e4",
+   "id": "50f135a5",
    "metadata": {
     "editable": true
    },
@@ -5286,7 +5264,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "68311282",
+   "id": "37f8c90c",
    "metadata": {
     "editable": true
    },
@@ -5298,7 +5276,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9730071c",
+   "id": "e87bd7c0",
    "metadata": {
     "editable": true
    },
@@ -5367,7 +5345,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f187ed7f",
+   "id": "93a0a150",
    "metadata": {
     "editable": true
    },
@@ -5381,7 +5359,7 @@
   {
    "cell_type": "code",
    "execution_count": 31,
-   "id": "371b1ef7",
+   "id": "ab36dde6",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -5394,7 +5372,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "32a3bda9",
+   "id": "37c5a3da",
    "metadata": {
     "editable": true
    },
@@ -5408,7 +5386,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "853e44bf",
+   "id": "28d6016e",
    "metadata": {
     "editable": true
    },
@@ -5421,7 +5399,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5257f382",
+   "id": "8dd5a626",
    "metadata": {
     "editable": true
    },
@@ -5432,7 +5410,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "26d17992",
+   "id": "672a58d0",
    "metadata": {
     "editable": true
    },
@@ -5444,7 +5422,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "54412ab6",
+   "id": "764d2bfe",
    "metadata": {
     "editable": true
    },
@@ -5454,7 +5432,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6ff6b804",
+   "id": "97882905",
    "metadata": {
     "editable": true
    },
@@ -5466,7 +5444,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1afe698d",
+   "id": "4bcfb4a5",
    "metadata": {
     "editable": true
    },
@@ -5482,7 +5460,7 @@
   {
    "cell_type": "code",
    "execution_count": 32,
-   "id": "ebbb3add",
+   "id": "b28165c1",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -5533,7 +5511,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "47ce5537",
+   "id": "d7309f97",
    "metadata": {
     "editable": true
    },
@@ -5543,7 +5521,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8de37a33",
+   "id": "e76cea5f",
    "metadata": {
     "editable": true
    },
@@ -5589,7 +5567,7 @@
   {
    "cell_type": "code",
    "execution_count": 33,
-   "id": "edc4ff78",
+   "id": "db0cf765",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -5602,7 +5580,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d85f3402",
+   "id": "b4e9ba3a",
    "metadata": {
     "editable": true
    },
@@ -5613,7 +5591,7 @@
   {
    "cell_type": "code",
    "execution_count": 34,
-   "id": "505a5d4e",
+   "id": "1f83c5ec",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -5628,7 +5606,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "720645ba",
+   "id": "ba510f9a",
    "metadata": {
     "editable": true
    },
@@ -5646,7 +5624,7 @@
   {
    "cell_type": "code",
    "execution_count": 35,
-   "id": "9575e845",
+   "id": "e9ff0303",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -5663,7 +5641,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7f3a7f52",
+   "id": "05274b95",
    "metadata": {
     "editable": true
    },
@@ -5673,7 +5651,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1c35ae76",
+   "id": "ba68440c",
    "metadata": {
     "editable": true
    },
@@ -5684,7 +5662,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "85e0db13",
+   "id": "a5adcaef",
    "metadata": {
     "editable": true
    },
@@ -5695,7 +5673,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "60c57a39",
+   "id": "5cc510cf",
    "metadata": {
     "editable": true
    },
@@ -5706,7 +5684,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a4f9e59a",
+   "id": "dbd377d2",
    "metadata": {
     "editable": true
    },
@@ -5729,7 +5707,7 @@
   {
    "cell_type": "code",
    "execution_count": 36,
-   "id": "a7420272",
+   "id": "6e3ddf8f",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -5742,7 +5720,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f2d73fbf",
+   "id": "3efb1da7",
    "metadata": {
     "editable": true
    },
@@ -5757,7 +5735,7 @@
   {
    "cell_type": "code",
    "execution_count": 37,
-   "id": "b6ff78b9",
+   "id": "05a87489",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -5837,7 +5815,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b3c3d94c",
+   "id": "abc50ad3",
    "metadata": {
     "editable": true
    },
@@ -5852,7 +5830,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7cb44deb",
+   "id": "177fafc6",
    "metadata": {
     "editable": true
    },
@@ -5865,7 +5843,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3ed162c2",
+   "id": "b29ed127",
    "metadata": {
     "editable": true
    },
@@ -5876,7 +5854,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e8ee4e88",
+   "id": "0b116bb9",
    "metadata": {
     "editable": true
    },
@@ -5888,7 +5866,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "548b53f1",
+   "id": "ea58b040",
    "metadata": {
     "editable": true
    },
@@ -5898,7 +5876,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "99e1b5e1",
+   "id": "87f83d2d",
    "metadata": {
     "editable": true
    },
@@ -5910,7 +5888,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5f21d11f",
+   "id": "1fce45c4",
    "metadata": {
     "editable": true
    },
@@ -5920,7 +5898,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "24418b17",
+   "id": "fd8fb0b0",
    "metadata": {
     "editable": true
    },
@@ -5942,7 +5920,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "305c547b",
+   "id": "349fc873",
    "metadata": {
     "editable": true
    },
@@ -5954,7 +5932,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1a181cc6",
+   "id": "0a604939",
    "metadata": {
     "editable": true
    },
@@ -5964,7 +5942,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1caa38cc",
+   "id": "723ed0f3",
    "metadata": {
     "editable": true
    },
@@ -5976,7 +5954,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cc8b8354",
+   "id": "9d257343",
    "metadata": {
     "editable": true
    },
@@ -5986,7 +5964,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "87fa8b39",
+   "id": "b2ad768a",
    "metadata": {
     "editable": true
    },
@@ -5998,7 +5976,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "de1323ec",
+   "id": "6dd8274f",
    "metadata": {
     "editable": true
    },
@@ -6011,7 +5989,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a34b0d9e",
+   "id": "1009148d",
    "metadata": {
     "editable": true
    },
@@ -6023,7 +6001,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9f857491",
+   "id": "64034477",
    "metadata": {
     "editable": true
    },
@@ -6033,7 +6011,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "05fc193f",
+   "id": "e2b9cd4e",
    "metadata": {
     "editable": true
    },
@@ -6045,7 +6023,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f4bcc6f5",
+   "id": "d0138e52",
    "metadata": {
     "editable": true
    },
@@ -6055,7 +6033,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9729c86a",
+   "id": "f5724eba",
    "metadata": {
     "editable": true
    },
@@ -6067,7 +6045,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "02c6b814",
+   "id": "9dd39529",
    "metadata": {
     "editable": true
    },
@@ -6077,7 +6055,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6e1350d4",
+   "id": "0a741a5f",
    "metadata": {
     "editable": true
    },
diff --git a/doc/src/week35/week35.do.txt b/doc/src/week35/week35.do.txt
index dbe808311..7bab8bb9c 100644
--- a/doc/src/week35/week35.do.txt
+++ b/doc/src/week35/week35.do.txt
@@ -1706,13 +1706,6 @@ and using the orthogonality of the matrix $\bm{U}$ we have
 !et
 We define $\bm{\Sigma}^T\bm{\Sigma}=\tilde{\bm{\Sigma}}^2$ which is  a diagonal matrix containing only the singular values squared. It has dimensionality $p \times p$.
 
-This means, using the orthogonality of $\bm{V}$,  that we get
-
-!bt
-\[
-\bm{X}^T\bm{X}=\tilde{\bm{\Sigma}}^2.
-\]
-!et
 
 We can now insert the result for the matrix $\bm{X}^T\bm{X}$ into our equation for ordinary least squares where
 
@@ -1725,10 +1718,10 @@ and using our SVD decomposition of $\bm{X}$ we have
 
 !bt
 \[
-\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{\Sigma}\bm{V}^T\tilde{\bm{\Sigma}}^{-2}\bm{V}\bm{\Sigma}^T\bm{U}^T\bm{y},
+\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{\Sigma}\bm{V}^T\left(\bm{V}\tilde{\bm{\Sigma}}^{2}(\bm{V}^T\right)^{-1}\bm{V}\bm{\Sigma}^T\bm{U}^T\bm{y},
 \]
 !et
-which gives us, using the orthogonality of the matrices $\bm{U}$ and $\bm{V}$,
+which gives us, using the orthogonality of the matrices $\bm{U}$ and $\bm{V}$,,
 
 !bt
 \[