small update

doc/LectureNotes/_build/html/week35.html

@@ -1802,15 +1802,15 @@ decomposition of the design matrix.</p>
 <p>which gives us, using the orthogonality of the matrix <span class="math notranslate nohighlight">\(\boldsymbol{V}\)</span>,</p>
 <div class="math notranslate nohighlight">
 \[
-\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
+\tilde{y}_{\mathrm{OLS}}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
 \]</div>
+<p>which is not the same as <span class="math notranslate nohighlight">\(\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}\)</span>, which due to the orthogonality of <span class="math notranslate nohighlight">\(\boldsymbol{U}\)</span> would have given us that the model equals the output.</p>
 <p>It means that the ordinary least square model (with the optimal
 parameters) <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}\)</span>, corresponds to an orthogonal
 transformation of the output (or target) vector <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> by the
 vectors of the matrix <span class="math notranslate nohighlight">\(\boldsymbol{U}\)</span>. <strong>Note that the summation ends at</strong>
 <span class="math notranslate nohighlight">\(p-1\)</span>, that is <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}\ne \boldsymbol{y}\)</span>. We can thus not use the
-orthogonality relation for the matrix <span class="math notranslate nohighlight">\(\boldsymbol{U}\)</span>. This can already be
-when we multiply the matrices <span class="math notranslate nohighlight">\(\boldsymbol{\Sigma}^T\boldsymbol{U}^T\)</span>.</p>
+orthogonality relation for the matrix <span class="math notranslate nohighlight">\(\boldsymbol{U}\)</span>.</p>
 </section>
 <section id="further-properties-important-for-our-analyses-later">
 <h2>Further properties (important for our analyses later)<a class="headerlink" href="#further-properties-important-for-our-analyses-later" title="Link to this heading">#</a></h2>