Consider the linear transformation T: R" R", and its associated matrix A =[T(ëi) T(ë»)T(em)] E R"Xm,At most, how many of the columns of A can be linearly independent?CLWDIIF8F6F7F9F10F11&789FILU OnO mOm+nOm -nOn - mO min(m, n)O max(m, n)W

Given linear transformation T: Rm -> Rn

Answered: Consider the linear transformation T:…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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From my linear algebra (ungraded) practice work: "Let M_n,n denote the vector space of n x n matrices, and let T: M_n,n --> M_n,n be the transformation that is given by T(A) = A + A^T... Prove that T is a linear transformation." I know how to show a transformation is general for a specific matrix or vector, but I'm not sure how to show that T is linear in general for an undefined nxn matrix?
Find the matrix representation of the linear operator T relative to B1. Then use Theorem 15 to find [T]B2
1. Let A = MB (f) be the matrix associated with the linear map f(x,y.z) = (3x-y+4z.y+z-x.x+y+z) relative to the bases B={u=(4,18,6), v= (17.1.0), w=(17,0,-5)} and B'= {u'=(2,2,1); v' = (1,0,0); w'=(3,2,0)) Then the first column of A is: OA-16 OB. 4 17 17 B 0 (3) 13 1 OC. 3 O D./ O E. 7-
11. If the matrix of a linear transformation T ::: F. (C) with respect to the ordered basis B = {(1, 0), (0, 1)} is i %3D i) Find the matrix of T with respect to ordered basis B' = {(1, 1), (1, – 1)} %3D
4. Let B = {v1, v2, V3} and C = {w1, w2, w3} be bases for R3, with vectors defined below. -(:) -() --() v1 = V2 = V3 = and --(:) --E) --() 1 wi = , w2 = 1 , W3 = 1 Let L: R³ → R³ be the linear transformation defined by L: Find [L]B and [L]c, the matrices associated to L with respect to B and with respect to C.
2. Let A = be an orthogonal 2 × 2 matrix. cos(0) sin(0 (a) Explain why the vector can be written as for some value of 0. a Use your answer to part (a) to find all 2×2 orthogonal matrices A = |C (b) (Hint: What can the second column of A look like?) (c) det(A) = 1? What kind of linear transformation do they correspond to? ! Given your answer to part (b), what are the 2 × 2 orthogonal matrices with
2. Let a = 2 ---- X= -2 4 and y = 4 2 (a) Find P, the projection matrix along the span of a. (b) Find proja (x). (c) Find proja (y). 3. Let A be a nonsingular 2 x 2 matrix. (a) Find P, the projection matrix onto C(A). (b) Find C(A) and comment on the result of part (a).
Suppose T is a linear transformation on R² (i.e. T(x, y) = (ax + by, cx + dy) where a, b, c, d are real numbers and, interpreting T as a matrix, det(T) # 0. Let P be the parallelogram which is the image of the square D = = [0, 1] × [0, 1] (i.e. T(D) =P). Then the area of P is given by O lad - bcl ad bc abcd 1 |ad - bc| O labcd|
Let B = { 1 + x, 1 − x², 1+x+x²} and B' = {2 − x, 1 − x +x²,3 + 2x}. 3.) Find the transition matrix from B to B'. If [9] B' = -3 4 what is g as a polynomial?
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Consider the linear transformation T: R" HR", and its associated matrix A = [T(ëi) T(ë») T(em)] E R"Xm, ... At most, how many of the columns of A can be linearly independent?