State with reasons whether each of the following maps S : Mat2.2(R) → Mat2.2(R) is linear. For1C ).1those maps that are linear: (i) find their matrix with respect to the basis0 0(* :)- (C )and (ii) find their rank and nullity.111(a) S(A) = A? (b) S(A) = CA, where C =2 4

Given: The mapping S:M2×2→M2×2 is defined as, a) S(A)=A2 b) S(A)=CA, where C=1224 To check: Whether…

Answered: State with reasons whether each of the…

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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5. Show that T is a linear transformation by finding a matrix that implements the mapping. Note that x₁, x2, and x3 are not vectors but are entries of a vector x in R³. T(x1, x2, x3) = (2x1 — x2, x1 − 2x2 + x3, x1 − 3x2 + x3) - X1
(a) Construct a linear map L(z)=az+b such that L(1)=i and L(i)=1 (b) Is the map you constructed in part (a) unique? Please Explain.
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.
1. (a) Define: invertible linear transformation. (b) T: (1,2) → (6x₁ − 8x2, −5x₁ +7x2) is a linear transformation from R2 to R². Show that T is invertible and find a formula for T-¹ alike to the one for T. Also, state the matrix representations for T and T-¹ in the standard bases.
Determine the matrix of the linear transformation T(x, y) = (3x + 5y, 3y – 2x) 5 -2 3 (a) -2 3 (b) (c) 3 3 3 (d) 3 3 -2 a 2]
Find the matrix of the linear transformation T from R? R?, where T([1, 0]) = [1, -2], and T([2, 1]) = [2, 3] (Note: matrix A = [T([1,0]), T([0, 1])] ). 1 (a) 1 (b) 1. (c) -2 1 -2 3 -2 1 (d) 0. 3
5. Consider a linear transformation T: R2 → R2 which reflects a vector about the line y=-x, and dilates the reflected vector by a factor of 2. Find the standard matrix for the linear transformation.
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).
8-3) show that T is a linear transformation by finding a matrix that implements the mapping. Note that x1, x2,... are not vectors but are entries in vectors a) T(x1, x2) = (2x2-3x1, x1-4x2, 0, x2) b) T(x1, x2, x3, x4) = 2x1 + 3x - 4x4 (T:R* → R)
9. Consider the following 2 x 2 matrix A and basis S of R²: A ¹ = [² %) and 5 = {[_¹₂] · [_³;]} The matrix A defines a linear operator on R². Find a matrix B that represents the mapping A relative to the basis S.
Let the linear map f:R³ → R² and h: R² R³ M(f) be the matrix associated with f and M (h) be the matrix associated with h relative to the standard bases respectively. Then which of the following is true? " A. M(h) is a 4x2 matrix, M(f) is a 2x3 matrix and M (hof) is a 4×3 matrix. B. M (foh) is a 4×3 matrix, M (f) is a 2x3 matrix and M(h) 4x2 matrix. C. Both M (f) and M(h) are invertible since the linear maps f and h have inverses. D. M (f) is a 3x2 matrix and M(h) is 2×4 matrix

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State with reasons whether each of the following maps S : Mat2.2(R) → Mat2.2(R) is linear. For (6 9)- (C ) those maps that are linear: (i) find their matrix with respect to the basis (* :) (C )> ; and (ii) find their rank and nullity. (a) S(A) = A² (b) S(A) = CA, where C = %3D 4