update

doc/BookChapters/linalg.do.txt

@@ -153,13 +153,13 @@ print(x)
 or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is
 !bc pycod
 import numpy as np
-x = np.log(np.array([4.0, 7.0, 8.0])
+x = np.log(np.array([4.0, 7.0, 8.0]))
 print(x)
 !ec
 To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the _itemsize_ functionality (the array $x$ is actually an object which inherits the functionalities defined in Numpy) as 
 !bc pycod
 import numpy as np
-x = np.log(np.array([4.0, 7.0, 8.0])
+x = np.log(np.array([4.0, 7.0, 8.0]))
 print(x.itemsize)
 !ec
 
@@ -305,7 +305,10 @@ print(f"This is the mean for each column:{np.mean(a, axis=0, keepdims=True,dtype
 print(f"This is the mean value for each row:{np.mean(a, axis=1, keepdims=True,dtype=np.float64)}")
 # We print then the mean value for each row by  setting keepdims=False
 print(f"This is the mean value for each row with keepdims false:{np.mean(a, axis=1, keepdims=False,dtype=np.float64)}")
+!ec
 
+Another useful function is the _ravel_ function, which returns a flattened array as shown in the example here.
+!bc pycod
 # Ravel return a contiguous flattened array.
 print(f"Flatten  the matrix:{np.ravel(a)}")
 # It is the same as reshaping the matrix into a one-dimensional array
@@ -726,6 +729,11 @@ and continue till we have solved all $n$ sets of linear equations.
 
 
 
-
+The calculation of the inverse here assumes that it actually
+exists. In many machine learning applications there may be strong
+linear dependencies among the various columns and/or rows. In our
+discussions of linear regression we will dive into the mathematics of
+the singular value decomposition, an algorithm which will allow us to calculate the so-called pseudo-inverse.
+These details will be presented in our linear regression chapter.