Consider a dataset $x$ with $n$ features represented by $p-$dimensional feature

vectors $,i=1,2,$dots,$n.$ Explain the concept of feature scaling and data

normalization in the context of machine learning algorithms such as logistic

regression. Formulate the problem of feature scaling and data normalization

as a linear transformation where each feature $j$ is transformed to $j^{\text{'}}$ with

minimum min$(j)$ and maximum max$(j).$ Show mathematically how the

Min$-$Max scaling technique transforms the features $j$ to normalized features

$j^{\text{'}}.$ Discuss the advantages and limitations of Min$-$Max scaling compared to

other normalization techniques, such as Z$-$score normalization.