M =Find the matrix M of the linear transformation T: R³ → R² given by(13)X2T=3x1- x₂ + (-9) x3-6x1 + 7x2

Answered: M = Find the matrix M of the linear…

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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Consider a linear transformation T from R³ to R2 for which 0 3 6 T ([]) - · ~ ([1]) - · · (8) - C] = [{}], T = 8 2 T 0 = 5 Find the matrix A of T. A = =
- R? defined by T(x, y) = (3x + 6y, 4x – y) and the base (24) Consider the linear transformation T: R² B = {(1,2), (3,4)} and B' = {(3,0), (0, –2)}. (a) Find As the standard matrix for T. (b) Find transition matrices PRB' and Pg'B- (c) Find Ags the matrix for T relative to B and the standard basis. (d) Find ARR! the matrix for T relative to B and B'. %3D (e) Suppose [ =La Find [T(5)]g,. (f) Find v and TT).
2. Write the quadratic function f (x,,y,)=-3x; +2x,x, – 3x; – 4x, – 4x, +6 in matrix form as f(x) = x'Ax + b'x + c and hence find af dx
1
-9 V₁ = --[-] and X₂ such that T(x) is Ax for each x. Let x = A = X₁ -2 [3] and let T: R² R² be a linear transformation that maps x into x₁v₁ +X₂V₂. Find a matrix A -8 and V₂ =
Let L be the linear transformation from R2 → R² given by L(x1, x2) = (2x1 - 3x2,4x1 + 4x2). (i) What is the matrix A of L? A = (ii) What is the determinant of L? Determinant = (iii) Is L shape preserving? (No answer given) (iv) Does L preserve orientation? (No answer given) (v) Is L injective? (No answer given) (vi) Is La clockwise rotation? (No answer given) (vii) Is L an anti-clockwise rotation? (No answer given) (viii) Compute the pre-image of the point (2, –1) under L.
(9) (?) (3) Let ei = and e2 = Let T be the linear transformation which sends e, to ( ) and sends e2 to (a) Find the image of ( -3 and X2 (b) What is the standard matrix of T? (4) Let T : R² → R² be the linear transformation given by r(:)-(, (:) xi + x2 4x1 + 5x2 T X2 3 Find x such that Tx = 8
Let f: R R be defined by f((x, y)) = 7y - 3x - 3. Is f a linear transformation? a. f((x₁, y₁) + (x₂, y₂)) = etc.) f((x₁, y₁>) + f((x₂, y₂)) = + Does f((x1, y₁) + (x2, Y2 )) = f ((x₁, y₁ )) + f((x2, Y₂ )) for all (x₁, y₁), (x2, Y₂) E R ²? choose b. f (c(x, y)) = c(f((x, y))) = Does f(c(x, y)) = c(f ((x, y))) for all c ER and all (x, y) ER ²? choose c. Is f a linear transformation? choose . (Enter x₁ as x1, <
Let B= ((1, 3), (-2,-2)) and 8' = ((-12, 0), (-4, 4)) be bases for R2, and let be the matrix for T: R2 R2 relative to B. (a) Find the transition matrix P from B' to B. p= (b) Use the matrices P and A to find [v] and [7(v)le, where [vla [-1 31. [v] = [T(v)18- 11 p-1 11 (c) Find pand A' (the matrix for 7 relative to 8). 41 41 (d) Find [7(vil bwo ways. (7)=P(7) [vila Al- 11 11
3x1-x2 (E) 1. Let T : R? → R³ be the linear transformation defined as T 2x1 + x2 %3D -X1 - X2 (a) Find its matrix representation. (b) Find T(0). (c) Find T-(0). 1 (d) Find T-
3x₁ - x₂ [-x₁ + x₂] 1. (i) Show that 7 ([x₂]) = [³ T (ii) Find the matrix representation is a linear transformation. of T relative to the standard basis.
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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M = Find the matrix M of the linear transformation T: R³ → R² given by X1 3x1 = T (1:)) - [³ x2 X3 - − x₂ + (−9) x3 -6x1 +7x₂