Question

5. Let B be the following matrix in reduced row-echelon form: 1 B= 1 -1 0-1 0 0 2 0 0 0 0 0 0 0 0 (a) (3 pts) Let C be a matrix with rref(C) = B. Find a basis of ker(C). (b) (3 pts) Find two matrices A1 and A2 so that rref(A1) = rref(A2) im(A) # im(A2). B, and 1 (c) (5 pts) Find the matrix A with the following properties: rref(A) = B, is an 0 eigenvector of A with eigenvalue 1, and is an eigenvector of A with eigenvalue 2. You can leave your answer as a product of matrices and inverse of matrices (you do not need to compute any inverse: if you need one, just leave it as -1 of a suitable matrix). For credit, you have to motivate your choice. Note that checking that the matrix you exhibit works does not provide any evidence as for why you chose the matrix. 6. Consider the following matrix A= (0 V2 1 2 0 (a) (6 pts) Determine the singular values of A. (b) (9 pts) Find the singular value decomposition of A. Your answer has to consist of three matrices U, E, V satisfying the appropriate properties and multiplied together to retrieve A. 7. (5 pts) Let T: R4 → R4 be the linear transformation with matrix 1 A= 2 2 0 0 -2 1 0 2 0 0 2 -2 0 2 2

Answer 1

B O ToOo OHOO T doo la) Let C be a matrix such that rref(c) = B (e) is the solntion set of cx = 0, where x = Then ker -8 Since ref (c) = B e is row equivalent to B and equivalent matrix als have the same solution set, So it is sufficient to solve the Systein By=0 Now By implies that -OOO līOOO OOO T doo 22 2 ) xq lie - } % Hz - xq x3 € 2 2q = 0 Since ist and 3rd Cohemns of B one pivotal Column. So, take H4= ti Then from o xz 224 - -27 R2 +84 & tt and 거 of + [1] - [*]-[1]• [] SS لا tt 2 where s, t are arbitrary constants So, a Basis of ker (c) is 1 - 2 1 Consider A, 0 - 1 and Az = B then Yeef (41) = yref (42) - B ont In Go ) = Span } } { [:],[! Jun (1.)Sve span { [:1, 1:11:13 whose cleanly I a vector in fm (AI) 3rd component i non Zeno (ac. [:]) But there does not exist a Im (A2) whose 3rd (because grd components are zero in each rector) So, Im (A.) & fm (A2). vector in Component r non-zeno Scanned with CamScanner