Suppose that V is a 3-dimensional vector space over a field F and T : V → V is a linear tion such that the corresponding F[x]-module structure on V is given by 7. V F[x]/(x3-x2-x + 1). Among the...
Let V be a finite-dimensional vector space, and let f :V + V be a linear map. Let also A be a matrix representation of f in some basis of V. As you know, any other matrix representation of f is similar to A. Show, conversely, that every matrix similar to A is a matrix representation of f with respect to some basis of V.
please help me with questions 1,2,3 1. Let V be a 2-dimensional vector space with basis X = {v1, v2}, write down the matrices [0]xx and [id]xx. 2. Let U, V, W be vector spaces and S:U +V, T:V + W be linear transforma- tions. Define the composition TOS:U + W by To S(u) = T(S(u)) for all u in U. a. Show that ToS is a linear transformation. b. Now suppose U is 1-dimensional with basis X {41}, V...
Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T : V -» V is matrix representation with respect to every basis of V. Prove that the dimension of linear transform ation that has the same that T must be a scalar multiple of the identity transformation. You can assume V is 3 Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T :...
Let T be a linear operator on a finite dimensional vector space with a matrix representation A = 1 1 0 0] 16 3 2 1-3 -1 0 a. (3 pts) Find the characteristic polynomial for A. b. (3 pts) Find the eigenvalues of A. C. (2 pts) Find the dimension of each eigenspace of A. d. (2 pts) Using part (c), explain why the operator T is diagonalizable. e. (3 pts) Find a matrix P and diagonal matrix D...
Question 1. Let V be a finite dimensional vector space over a field F and let W be a subspace of Prove that the quotient space V/W is finite dimensional and dimr(V/IV) = dimF(V) _ dimF(W). Hint l. Start with a basis A = {wi, . . . , w,n} for W and extend it to a basis B = {wi , . . . , wm, V1 , . . . , va) for V. Hint 2. Our goal...
1 point) Read 'Diagonalization Changing to a Basis of Eigenvectors' before attempting this problem. Suppose that V is a 5-dimensional vector space. Let S -(vi,... , vs) be some ordered basis of V, and let T-(wi.... . ws) be some other ordered basis of V. Let L: V → V be a linear transformation. Let M be the matrix of L in the basis Sand et N be the matrix of L in the basis T. Decide whether each of...
Let V be a vector space over a field F, and let U and W be finite dimensional subspaces of V. Consider the four subspaces X1 = U, X2 = W, X3 = U+W, X4 = UnW. Determine if dim X; <dim X, or dim X, dim X, or neither, must hold for every choice of i, j = 1,2,3,4. Prove your answers.
1. Let T be the matrix T=10 3 acting on the complex vector space V C3 (a) Recall how T defines the structure of a C-module on C3. (b) Let p(x71, and let 2Compute the element p(x) v of C3 (c) Give a set of generators and relations for C3 over Cz] with the above module structure. (d) Write down the relations matrix (e) Recall the definition of minimal polynomial of a matrix. (f) What is the minimal polynomial of...
How can I proof if V a finite dimensional vector space over a field, and T is a linear transformation V->V, V is cyclic relative to T iff minimal polynomial of T is equal to the characteristic polynomial of T?
Problem 9. Let V be a vector space over a field F (a) The empty set is a subset of V. Is a subspace of V? Is linearly dependent or independent? Prove your claims. (b) Prove that the set Z O is a subspace of V. Find a basis for Z and the dimension of Z (c) Prove that there is a unique linear map T: Z → Z. Find the matrix representing this linear map and the determinant of...