MATH 218 Final, Fall 2002

7 nN A N AUB eel Mathematics 218 BRAY y Final Exam Fall: 2002-2003 : Time: 2h. 2 3 3 1) Calculate A and A and show that A 2) Deduce that A is invertible and Find A ( 10pts ) 1) Use the rank to determine the values of & for which the linear system \ xtky= 1 | kx+ y=-l has no solution, one solution or infinitely many solutions. 2) Fork=1, find the least square solution(s) of the linear system ( 15pts ) . 1 4 5 III- Let V= R° be the Euclidean inner product and A=| 2 1 3 3 2 -1 Ky and let W be the Null space of A. ara 1) Findrank( A). 2) Find a basis for W and a basis for W+ 3)Let v = (1,1,1) . Find the orthogonal projection of von W . ( 15pts ) ~ Let V = Maxn_ be the vector space of all nxn matrices. Let C be a fixed nxn invertible matrix and T:V > V _ be 1) Show that T isa linear operator 2) Show that Tis one-to-one and deduce rank(T) = on 2 a \ the function defined by T(A)=CA for all A in Maxa \ 1 V- Let A= 1 1 2 1) Show that the characteristic equation of A is A (A -3) =0 2) Find an orthonormal basis of every eigenspace corresponding to every eigenvalue 3) Find a matrix P that orthogonally diagonalizes A. ( 20pts ) vi VI- Prove or disprove (by a counter example) the following: 1) Every square matrix with a non zero determinant has a QR-decomposition. 2) Let A, B be row equivalent nxn matrices. a) There exists an invertible nxn matrix C suchthat B=CA b) A is invertible if and only if B is invertible c) A and B have the same Null space. d) A and B have the same column space VIT- Let V be an inner product space and let T:V > V be a linear operator. Suppose that ||T(v){| = ||vi} for all v in V. 1) Show that T is one-to-one . 2) Show that < T(u), T(v) >= <u,v> for all u,v in V ( Hint: Use ||T(ut+v)]]? = Jfutvil? ) 3) Suppose dim( V)=2 and let B={ vj, v2 } be an orthonormal basisof V . Let Bi= {T(vi) , T( v2) } a)Show that B, is an orthonormal set of V b) Show that B, is a linearly independent set of V. c) Deduce that B, is an orthonormal basis of V.