for the smoothing cubic spline of section show that (xumax{ , } min{ ,umin{ , } let be an model matrix and let be the unit-length vector with -th entry equal to one (uk kuk suppose that the -th column of is and that it is replaced with new predictor wso that we obtain the new model matrixe ( )ux (adenoting : ( vkw vk show that ex xx udduxx ( )( ( )( dx ex differs from xx by symmetric matrix of rank two in other wordsx (bsuppose that :(xx gi )- is already computed explain how the sherman-morrison formulas in theorem can be applied twice to comex gi in (( )ppute the inverse and log-determinant of the matrix computing timerather than the usual (( )pcomputing time (cwrite python program for updating matrix (xx gi )- when we change the -th column of xas shown in the following pseudo-code algorithm updating via sherman-morrison formula inputmatrices and bindex kand replacement for the -th column of outputupdated matrices and set to be the -th column of set to be the unit-length vector such that uk kuk budb dbu bdub ubd update the -th column of with return xb use algorithm from exercise to write python code that computes the ridge regression coefficient in ( and use it to replicate the results on figure the following pseudo-code (with running cost of (( ) )may help with the writing of the python code this sherman-morrison updating is not always numerically stable more numerically stable method will perform two consecutive rank-one updates of the cholesky decomposition of xx gi