4. For a given angle k, the matrix is a rotation matrix. That is, for a given vector direction. E R2, Gr is just the vector rotated by in the clockwise (a) Consider a matrix H of size m à n, and suppose the (2 x 1) submatrix he has its lower entry nonzero. Construct an mx m unitary matrix Im,, of the form 0 where Gk is a matrix of the forzu (2) and Ip is the p à p identity matrix, so that B-r,n,kH has the submatri:x l,j Such a unitary matrixãmk is called a Givens Rotation. Explicitly give formulas for the entries of Gk and the value of in terms of the entrieshj and h+1j. (b) In the GMRES algorithm in step n, a QR factorization of nmust be computed, where Hn is Ilessenberg and of size (n+1) xn. However, from the previous step a QR factorization of f-1 is already known, where HT-1 is the first n rows and first n-1 columns of Hn. Given where c is an n x 1 vector and a is a scalar, and given the QR factorizationQn-1R-1 where Qn-i has been constructed by a sequence of n -1 Givens rotations, describe how to construct a QR factorization of H, Hn-QnR in O(n) operations using n-1 and one additional Givens rotation, and show how Qn and Rn are related to -1 and Rn-1