MTH102N ASSIGNMENTLA 5
(1) Let C be an m n matrix and let T : Rn Rm be the linear transformation dened by C . Show that the matrix of T with respect to the standard bases of Rn and Rm is C . (2) Let the linear map T : R2 R2 be given by T (x, y ) = (ax + by, cx + dy ). Find the matrix of T with respect to the standard basis of R2 . Now do the same by considering the basis {(0, 1), (1, 0)} on domain and range of T . (3) Consider the linear map T : C C given by T (z ) = iz. By considering the basis {1, i} of C (over R) on domain and codomain of T nd the matrix of T. (4) Let T : V V be a linear map such that Ker(T ) = Range(T ). What can you say about T 2 . On R2 can you give example of such a map? (5) Does there exist a linear transformation T : R2 R4 such that Range(T ) = {(x1 , x2 , x3 , x4 ) : x1 + x2 + x3 + x4 = 0}? (6) Let V be a vector space of dimension n and let A = {v1 , . . . , vn } be an ordered basis of V . Suppose w1 , . . . , wn V and let (a1j , . . . , anj )t denote the coordinates of wj with respect to A. Put C = [aij ]. Then show that w1 , . . . , wn is a basis of V if and only if C is invertible. (7) Find the range and kernel of T : Rn Rn given by T (x, y, z ) = (x + z, x + y + 2z, 2x + y + 3z ). (8) Let T be a linear transformation from an n dimensional vector space V to an m dimensional vector space W and let C be the matrix of T with respect to a basis A of V and B of W . Show that (a) (T ) = rank(C ); (b) T is one-one if and only if rank(C ) = n; (c) T is onto if and only if rank(C ) = m; (d) T is an isomorphism (that is, one-one and onto) if and only if m = rank(C ) = n. (9) Let <, > be any inner product on Rn . Show that < x, y >= xt Ay for all vectors x, y Rn where A is the symmetric n n matrix whose (i, j )th entry is < ei , ej >, the vector ei being the standard basis vectors of Rn . (10) Show that the norm of a vector in a vector space V has the following three properties (a) v 0 and v = 0 if and only if v = 0.
1
MTH102N ASSIGNMENTLA 5
(b) v = || v for all R and v V . (c) v + w v + w for all v, w V . (11) Use Gram-Schmidt process to transform each of the following into an orthonormal basis; (a) {(1, 1, 1), (1, 0, 1), (0, 1, 2)} for R3 with dot product. (b) Same set as in (i) but using the inner product dened by < (x, y, z ), (x , y , z ) >= xx + 2yy + 3zz . (12) Let U be a proper subspace of the inner product space V . Let U = {v V : < v, u >= 0 u U }. Show that U is a subspace of V ( it is called orthogonal complement of U ). Let U = {(1, 2, 3) : R} be a subspace of R3 with scalar product. Find U . Also, show that S is a subspace of V for any arbitary subset S of V. (13) Let U1 and U2 be subspaces of a vector space V . We say that V is the direct sum of U1 and U2 , notation V = U1 U2 , provided that each element of V has a unique expression in the form of v = x + y where x U1 and y U2 . (a) Show that V = U1 U2 if and only if U1 U2 = {0} and each element of V is expressible in the form v = x + y where x U1 and y U2 . (b) Show that V = U U for any subspace U of the inner product space V . (14) Let Rn and Rm be equipped with usual dot product and let A be an m n matrix with real entries. Show that Ker A = (Im At ) and Im A = (Ker At ) . (15) Let A be an n n matrix and b be a column vector in Rn . Let x = (xi ) be a column vectors of unknowns. Use the previous problem to show that only one of the following can have a solution for x (i) Ax = b (ii) At x = 0 and xt b = 0 (This is referred as Fredholm Alternative) (16) Let A be an n n real matrix. Show that the following are equivalent (a) A is orthogonal. (b) A preserves length, i.e. Av = v (c) A is invertible and At = A1 . (d) The rows of A forms and orthonormal basis of Rn . (e) The columns of A forms an orthonormal basis of Rn . v Rn .
