Let T: R² R² be defined by T-{[5].[3]}C=Given Pc =[[T](PB (u))12=3x1x2-3([2₂])-[¹22] Lu= [2] B-{].G]}, and[3]}=Let-x1 +Ex: 5x1=-2, use the Fundamental Theorem of Matrix Representations to find [T](PB(u)).5

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Answered: Let T: R² → R² be defined by 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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Let A be a 2 × 2 matrix, and let LA be the linear operator defined by LA(x) = Ax Show that (a) LA maps R2 onto the column space of A. (b) if A is nonsingular, then LA maps R2 onto R2.
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Consider the operator R2 → R2given by left-multiplication by the matrix: B =[6 −7, 1 −2]. Find the characteristic polynomial cB(x), the eigenvalues of B, and theeigenspaces of B.
Let A be an n x n matrix over C. Let f be an entire function, i.e. an analytic function on the whole complex plane, for example exp(z), sin(z), cos(z). Using the Cayley-Hamilton theorem we can write ƒ(A) = an−1A”−¹ + an−2A¹−²+...+ a₂Ã² + a₁A+aoIn (1) where the complex numbers ao, a₁, ..., an-1 are determined as follows: Let r(λ) := An−1\”−¹ + an-21²-2+...+ a₂λ² + a₁λ + ao which is the right-hand side of (1) with A¹ replaced by X³, where j = 0, 1, ….., n— 1 of each distinct eigenvalue λ; of the matrix A, we consider the equation ƒ(λj) =r(λj). (2) If X, is an eigenvalue of multiplicity k, for k> 1, then we consider also the following equations ƒ'(A)|a=a; = r′(A)|a=a; ƒ"(A)|a=a; = r″(A)|a=a; ...... f(k-1) (X) = (k-1)(A) (A)|x 1x=x₂ Apply this technique to find exp(A) with с A = = (ε c), C C * |x=x₂ CER, C# 0.
Let f(u, v) = (u - v, 1, u + v) and g(x, y, z) = xyz. Find the entry a12 (i.e. the entry in the first row and second column) of the derivative matrices Df(u, v), Dg(x, y, z) and D(gof)(0, 1). (a) a 12 of Df(u, v) is (b) a12 of Dg(x, y, z) is (c) a12 of D(gof)(0, 1) is (d) Select the correct answer about D(g. f)(u, v) D(gof)(u, v) is a 1 x 2 matrix. D(gof)(u, v) is a 2 x 2 matrix. D(gof)(u, v) is a 3 x 2 matrix. D(gof)(u, v) is a 2 x 3 matrix. D(gof)(u, v) is a real-valued function of u and v. (e) Select the correct answer about D(f g)(x, y, z) OD(fog)(x, y, z) is not defined. D(fog)(x, y, z) is a real-valued function of x and y. D(fog)(x, y, z) is a 3 x 2 matrix. D(fog)(x, y, z) is a 2 x 2 matrix. D(fog)(x, y, z) is a 2 x 1 matrix.
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Let T: R² → R² be defined by T = {[5].[a]} C= Given Pc 1 -2 = [223] -2 5 [T](PB (u)) = Ex: 5 ([2/2]) = 3x1 x2 -x₁ + x₂. . Let u = ={PD-R]} B= , and , use the Fundamental Theorem of Matrix Representations to find [T] (PÂ(u)).