$(40$ pts$)$ Problem $3$

Consider the following polynomial set

$($i$)(20$ pts$)$ For a given large data set $(x_i,y_i),(i=1$:$n),n1,$ you are to derive the matrix

problem using linear least$-$squares regression for the best fit of the form

$y(x)={\displaystyle \underset{j=0}{\overset{k}{}}}{a}_j{T}_j(x)$

You must show all steps in your derivation. Hint: You may use $T_{ji}=T_j(x_i)$ for shorthand nota$-$

tion. Do not attempt to solve the matrix problem.