show thatfor the squared-error lossthe approximation error `(gg `(gin ( )is equal to (gg (xg( )) [hintexpand `(gg ( (xg(xgg ( )) suppose is the class of linear functions linear function evaluated at feature can be described as (xbx for some parameter vector of appropriate dimension denote gg (xxbg and ggt (xx> show that    ggt (xg(xe > xbg xbg (xhencededuce that the statistical error in ( is `(ggt `(gg (ggt (xgg ( )) show that formula ( holds for the - loss with - response let be an -dimensional normal random vector with mean vector and covariance matrix swhere the determinant of is non-zero show that has joint probability density - ( -us ( -uf (xn ( |sx rn let ay using the defining properties of the pseudo-inverseshow that for any rpkab yk kab yk suppose that in the polynomial regression example we select the linear class of functions with thengg and the approximation error is zerobecause gg (xg(xxbwhere [ - - ] use the tower property to show that the learner gt (xx> with xyassuming rank( is unbiasede gt (xg( unbiased (exercise continued observe that the learner gt can be written as linear combination of the response variablegt (xxxy prove that for any learner of the form xaywhere pxn is some matrix and that satisfies ex [xayg( )we have varx [xxy varx [xay]where the equality is achieved for xthis is called the gauss-markov inequality henceusing the gauss-markov inequality deduce that for the unconditional variancegauss-markov inequality var gt ( var[xaydeduce that xalso minimizes the expected generalization risk consider again the polynomial regression example use the fact that ex xh( )where (ue[ [ ( ) (un )]to show that the expected in-sample risk iskh( ) kxxh( ) ` ex `in (gt alsouse theorem to show that the expected statistical error isex ( ) ( `tr( ( ) (xh(ub) (xh(ub