Question $1.(20$ pts$.)$ Given a training set with $m$ examples, the standard form of the

regularized loss function for linear regression is

$J()=\frac{1}{2m}[{\displaystyle \underset{i=1}{\overset{m}{}}}(h_{}(x^{(i)})-y^{(i)}{)}^2+{\displaystyle \underset{j=1}{\overset{n}{}}}{}_j^2]$

Also, consider the following loss function which is in canonical form

$J^{\text{'}}()=(Y-x{)}^T(Y-x)+{}^T$

where features are represented as a matrix $x($each row is an example and each

column is an individual feature$)$ and true target values are represented as a column

vector $Y.$

$($a$)$ Are the two functions $J()$ and $J^{\text{'}}()$ exactly equivalent? Explain briefly.

$($b$)$ Suppose we accidentally write $J^{\text{'}}()=(Y-x{)}^T(Y-x)+Y^{T}Y$ instead.

Explain briefly what happens with this form of "regularization".

$($c$)$ Suppose we use the correct expression but accidentally choose Explain

briefly how this defeats the purpose of regularization.