Prove or disprove (8) Let T: V → V be linear. If T is diagonalizable, then...
Let V and W be a vector spaces over F and T ∈ L(V, W) be invertible. Prove that T-1 is also linear map from W to V . Please show all steps, thank you
Let and let T : R4 → R4 be the map T(v) = v-2 proju,V-2 proju, w (a) Show that T is a linear transformation. (b) Find T] and (c) Show that T is invertible and find T-1 Let and let T : R4 → R4 be the map T(v) = v-2 proju,V-2 proju, w (a) Show that T is a linear transformation. (b) Find T] and (c) Show that T is invertible and find T-1
1. Let V be a vector space with bases B and C. Suppose that T:V V is a linear map with matrix representations Ms(T)A and Me(T) B. Prove the following (a) T is one-to-one iff A is one-to-one. (b) λ is an eigenvalue of T iff λ is an eigenvalue of B. Consequently, A and B have the same eigenvalues (c) There exists an invertible matrix V such that A-V-BV 1. Let V be a vector space with bases B...
mathematics or linear algebra. Let AA be an m×nm×n matrix. Prove that AATAAT is orthogonally diagonalizable.
Let T be a linear operator on F2. Prove that if v f 0 is not an eigenvector for T, then v is a cyclic vector for T. Conclude that either T has a cyclic vector T is a scalar multiple of the identity.
Let T: V + W be a linear transformation. Assume that T is one-to-one. Prove that if {V1, V2, V3} C V is a linearly independent subset of V, then {T(01), T(v2), T(13)} C W is a linearly independent subset of W.
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> V be a linear transformation (a) Prove that for every pair of ordered bases B = (Ti,...,T,) of V and C = (Wi, ..., Wm) of W, then exists a unique (B, C)-matrix of T, written A = c[T]g. (b) For each n e N, let Pn be the vector space of polynomials of degree at mostn in the...
I need the answer to problem 6 Clear and step by step please Problem 4. Let V be a vector space and let T : V → V and U : V → V be two linear transforinations 1. Show that. TU is also a linear transformation. 2. Show that aT is a linear transformation for any scalar a. 3. Suppose that T is invertible. Show that T-1 is also a linear transformation. Problem 5. Let T : R3 →...
7.) 10points Let V be the space of 2 x 2 matrices. Let T: V-V be given by T(A) = A a.) Prove that T a linear transformation b.) Find a basis for the nullspace (Kernel) of T. c) Find a basis for the range of T. 7.) 10points Let V be the space of 2 x 2 matrices. Let T: V-V be given by T(A) = A a.) Prove that T a linear transformation b.) Find a basis for...
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> W be a linear transformation. (a) Prove that for every pair of ordered bases B = exists a unique m x n matrix A such that [T(E)]c = A[r3 for all e V. The matrix A is called the (B,C)-matrix of T, written A = c[T]b. (b) For each n E N, let Pm be the vector space of...