4. Let B = {v1, V2, V3} and C = {w1, w2, w3} be bases for R, with vectors defined below.(:)V1 =-3• U2 =• V3 =1and1wi =w2 =w3 =-1Let L: R3 → R³ be the linear transformation defined by(:) - ()yFind [L]B and [L]c, the matrices associated to L with respect to B and with respect to C.

Introduction: A linear transformation is a characteristic from one vector area to every other that…

Answered: 4. Let B = {v1, v2, V3} and C = {w1,w2,…

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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Show that T is a linear transformation by finding a matrix that implements the mapping. Note that X₁, X2, vectors but are entries in a vector. A = T(X₁ X₂ X3 X4) = 3x₁ + 4x₂ − 3x3 + x4 X2 (T: R→R) are not
a b Let M2 x2 be the vector space of all 2 x2 matrices, and define T: M2×2→M2×2 by T(A) = A+A', where A = c d a. Show that T is a linear transformation. b. Let B be any element of M2 x2 such that BT = B. Find an A in M, x2 such that T(A) = B. c. Show that the range of T is the set of B in M2 x2 with the property that B' = B. d. Describe the kernel of T.
7. Let T: R³ R³ be defined by (a) (b) (E) x1 + x2 + x3 = X1 X2 X3 [X1X2 X3] Find the matrix of linear transformation for T. T Find all vectors 7 € R³ such that T(x) = x.
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.
5. Find the orthogonal projection of b = (1,-1,2) onto the left null space of the matrix A = -6
Find a preimage of the vector 4 V= under the linear transformation T given by the matrix -3] = [44] -4 14 That is find the vector u such that T(u) = v. A U₁ ū₂
14. Find the matrix for the 2 R², transformation T: R² where T represents the transformation by reflecting a vector over the x axis, rotating a vector through an angle of 5л 6 ●
C: Let T be the linear operator associated in the standard basis B of F2 with the matrix 2 - [44] - 1 Let B' be the basis ((2, -1), (2, 1)) M = b) Using eachof these three matrices find the dimension of the range of T (you must check that the dimension is the same with all the matrices)

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4. Let B = {v1, v2, V3} and C = {w1,w2, w3} be bases for R³, with vectors defined below. -3 V2 = V3 = and 1 wi = w2 = w3 = 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.