Problem 5 Let U be an n dimensional vector space and T E L(U,U). Let I...
Let V be a finite-dimensional inner product space, and let U and W be subspaces of V. Denote dim(V) = n, dim(U) = r, dim(W) = s. Recall that the proj and perp maps with respect to any subspace of V are linear transformations from V to V. Select all statements that are true. Note that not all definitions above may be used in the statements below If proju and perpu are both surjective, then n > 0 If perpw...
QUESTION 5 Let V denote an arbitrary finite-dimensional vector space with dimension n E N Let B = {bi, bn} and B' = { bị, b, } denote two bases for V and let PB-B, be the transition matrix from B to B' Prove that where 1 V → V is the identity transformation, i e 1(v) v for all v E V Note that I s a linear transformation 14] QUESTION 5 Let V denote an arbitrary finite-dimensional vector...
7.3.1 Let U be a finite-dimensional vector space over a field F and T є L(U). Assume that λ0 E F is an eigenvalue of T and consider the eigenspace Eo N(T-/) associated with o. Let. uk] be a basis of Evo and extend it to obtain a basis of U, say B = {"l, . . . , uk, ul, . . . ,叨. Show that, using the matrix representation of T with respect to the basis B, the...
Problem 5 (25 points). Let Mat2x2(R) be the vector space of 2 x 2 matrices with real entries. Recall that (1 0.0 1.000.00 "100'00' (1 001) is the standard basis of Mat2x2(R). Define a transformation T : Mat2x2(R) + R2 by the rule la-36 c+ 3d - (1) (5 points) Show that T is linear. (2) (5 points) Compute the matrix of T with respect to the standard basis in Mat2x2 (R) and R”. Show your work. An answer with...
Problem 4. Let n E N, and let V be an n-dimensional vector space. Let(, ,): V × V → R be an nner product on V (a) Prove that there exists an isomorphism T: V -R" such that (b) Is the isomorphism T you found in part (a) unique? Give a proof or a counterexample. (c) Let A be an n × n symmetric matrix such that T A > 0 for all nonzero ERT. Show that there exists...
15 5. Let P2 and Pz denote the vector space of polynomials of degrees atmost 2 and 3 respectively. Let T:P2 P3 be the transformation that maps a polynomial p(t) to the polynomial (t - 2)p(t). (a) Find the image of p(t) = t2, that is, find T(t2). (b) Show that T is a linear transformation. (c) Find the matrix of T relative to the bases B = {1,t, tº} and C = {1,t, t², tº}. (d) Is T onto?...
Let V be a finite-dimensional vector space and let T L(V) be an operator. In this problem you show that there is a nonzero polynomial such that p(T) = 0. (a) What is 0 in this context? A polynomial? A linear map? An element of V? (b) Define by . Prove that is a linear map. (c) Prove that if where V is infinite-dimensional and W is finite-dimensional, then S cannot be injective. (d) Use the preceding parts to prove...
(d) (4 points) Let T : R² + Rº be the transformation that rotates any vector 90 degrees counterclockwise. Let A be the standard matrix for T. Is A diagonalizable over R? What about over C? (e) (3 points) Let T : R4 → R4 be given by T(x) = Ax, A = 3 -1 7 12 0 0 0 4 0 0 5 4 0 4 2 1 Is E Im(T)? 3 (f) (9 points) Let U be a...
Problem 2. For each polynomial p(t) = do +at+...+ amtm with real number coefficients and for each n x n matrix A, we define the n x n matrix p(A) by P(A) = ao In + a A+ ... + amA”. Also, for each n, let Onxn E Rnxn be the n x n zero matrix. (a) Show that for all polynomials p and q and square matrices A, we have p(A)q(A) = 9(A)p(A). (b) Show that for every 2...
Problem 4. Give an example of a linear operator T on a finite-dimensional vector space such that T is not nilpotent, but zero is the only eigenvalue of T. Characterize all such operators. Problem 5. Let A be an n × n matrix whose characteristic polynomial splits, γ be a cycle of generalized eigenvectors corresponding to an eigenvalue λ, and W be the subspace spanned by γ. Define γ′ to be the ordered set obtained from γ by reversing the...