1. Derivation of the Normal Equations: In problem 1, you determined the best linear model for a single variable; for this problem you will find the best linear model for multi- linear regression. Consider the problem of K input variables such that the input vector is given by x=[x1,x2,xK1,xK]T. Assume we want to fit a multi-linear model (without an intercept) of the form y^=1x1+2x2+3x3+KxK=Tx, where =[1,2,K1,K]T. Assume we have N measurements of the input and output where the n-th measurement is given by x(n) and y(n) with corresponding estimate y^(n)=Tx(n). We will also define the output measurement vector as y=[y1,y2,yN1,yN]T. To find the best model, we want to minimize the means square error (MSE), MSE=N1n=1N(y(n)y^(n))2 a) Show that the vector y^=[y^1,y^2,y^N1,y^N]T, can be written as y^=X, where X is a matrix. Explicitly show the rows of the matrix X. b) Show that the MSE can be written as MSE=yX2 where the "| "operator corresponds to the magnitude of a vector. c) Find an expression for the optimum model parameter vector . Hint*: Use the fact that the gradient of the MSE is given by yX2=2XTy+2XTX