3. Consider the vector space V = R2[x] with its standard ordered basisE = 1,x,x2
and the linear map
T :R2[x]−→R2[x], T(p)=p(x−1)−p(0)x2
(a) (1 point) What is [T]E?
(b) (1 point) Is T invertible?
(c) (6 points) Compute the eigenvalues of T and their algebraic multiplicity.
(d) (2 points) Is T diagonalisable? If so, find a matrix Q such that Q−1[T]EQ is diagonal. If not, findQ, so that the above matrix is upper triangular.
3. Consider the vector space V = R2[x] with its standard ordered basisE = 1,x,x2 and the linear map T :R2[x]−→R2[x], T(p...
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...
Font Styles Paragraph Definition 1: Given La linear transformation from a vector space V into itself, we say that is diagonalizable iff there exists a basis S relevant to which can be represented by a diagonal matrix D. Definition 2: If the matrix A represents the linear transformation L with respect to the basis S, then the eigenvalues of L are the eigenvalues of the matrix A. I Definition 3: If the matrix A represents the linear transformation L with...
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...
2. Let T: P(R) + P(R) be such that Tp(x) = P(1)x2 +p(1)+ p0). a) Show that T is a linear operator. b) Find a basis for Ker(T) and a basis for Range(T). c) Is T invertible? Why? d) If possible find a basis for P(R) such that [T], is a diagonal matrix. e) Find the eigenvalues and eigenvectors of S=T* - 31.
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...
(1 point) The linear transformation T: R4 R4 below is diagonalizable. T(x,y,z,w) = (x – - (2x + y), -z, 2 – 3w Compute the following. (Click to open and close sections below). (A) Characteristic Polynomial Compute the characteristic polynomial (as a function of t). A(t) = (B) Roots and Multiplicities Find the roots of A(t) and their algebraic multiplicities. Root Multiplicity t= t= t= t= (Leave any unneeded answer spaces blank.) (C) Eigenvalues and Eigenspaces Find the eigenvalues and...
Please provide answer in neat handwriting. Thank you Let P2 be the vector space of all polynomials with degree at most 2, and B be the basis {1,T,T*). T(p(x))-p(kr); thus, Consider the linear operator T : P) → given by where k 0 is a parameter (a) Find the matrix Tg,b representing T in the basis B (b) Verify whether T is one-to-one and whether or not it is onto. (c) Find the eigenvalues and the corresponding eigenspaces of the...
Let T be a linear map from R3[z] to R2[z] defined as (T p)(z) = p'(z). Find the matrix of T in the basis: 4 points] Let T be a linear map from Rals] to R12] defined as (TP)(z) = p,(z). Find the matrix of T in the basis: in R2[-]; ~ _ s, r2(z) (z-s)2 in R2 [2], where t and 8 are real numbers. T1(2 Find coordinates of Tp in the basis lo, 1, 12 (if p is...
For each of the following operators T on a vector space V, find an ordered basis B such that [T]e is a diagonal matrix. (a) V = P2 (R) and T(f(x)) = xf'(x) + f(2)x+ f(3). db (b) V = M2x2(R) and T b (1 :]) = [.. (c) V = M2x2(R) and T(A) = AT + 2tr(A)12.
4. Consider the following matrix [1 0 -27 A=000 L-2 0 4] (a) (3 points) Find the characteristic polynomial of A. (b) (4 points) Find the eigenvalues of A. Give the algebraic multiplicity of each eigenvalue (c) (8 points) Find the eigenvectors corresponding to the eigenvalues found in part (b). (d) (4 points) Give a diagonal matrix D and an invertible matrix P such that A = PDP-1 (e) (6 points) Compute P-and verify that A= PDP- (show your steps).