13. Let 4 : M₂(R) → R defined by p(d)) = a + d. Recall that the operation on M₂(R) is the usual additionCof matrices. Determine, with proof, if is a group homomorphism. If it is, find its Kernel.

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Answered: 13. Let : M₂ (R) → R defined by ( oll c…

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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Suppose that L1 : V → W and L2 : W → Z are linear transformations and E, F, and G are ordered bases for V, W, and Z, respectively. Show that, if A represents L1 relative to E and F and B represents L2 relative to F and G, then the matrix C = BA represents L2 ◦ L1: V → Z relative to E and G. Hint: Show that BA[v]E = [(L2 ◦ L1)(v)]G for all v ∈ V.
Which of the following properties are true for the transformation T : R³ → R² with (E)- -y x + y (a) T is one to one (b) T is onto (c) T is linear (b) and (c) (a), (b), and (c) (a) and (b) (c) only OO O
2. Let d≥ 1, let F be a field and let Affa(F) denote the set of all functions TA, FdFd which have the form T₁,v(x) = Ax + v, where A is an invertible d x d matrix with entries in F and where v Fd. Prove that Affa(F) is a group (where the binary operation is given by composition of functions).
2. Let R be the commutative ring of all matrices with rational entries, M3(Q), of the form a b c 0 ab 00 a Show that the mapping : R Q given by ab c 0 ab 00a is a homomorphism. $ - a
. Linear transformation T: Rm →R" is given by one of the matrices below. Find m and n. 1 (a) [ 1 4 ] [ (b) 4 -2 3 8 i] 0 11 (c) 2-2 3 -3
For the linear transformation L:R+R defined by L(x4, x2, x3) = (x1-X2, X2 -X3, X3-X4), find a basis for Range(L). Find rank(L). Note: Write down your answer in details into the below empty field.
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For a linear transformation L : R² → R³ is L(2, 1) = (2, 6, 8) and L(0, 1) = (2,0, 4). a) What is L(1,0)? b) What is the matrix representation of L? c) Use the matrix representation to calculate L(3,-1).
4. You are given a linear transformation f : R² → R² such that (H) - [-]- f 2 (a) for f. If not, explain why this is impossible. (b) assume F is such an orthogonal projection and find a possible matrix for f. If not, explain why this is impossible. (c) a reflection and find a possible matrix for f. If not, explain why this is impossible. Could f be a rotation? If so, assume f is a rotation and find a possible matrix Could f be an orthogonal projection onto a line through the origin? If so, Could f be a reflection about a line through the origin? If so, assume f is such 5. State whether the following are True or False. Each correct answer will receive With a valid explanation, a correct answer will receive (a) For any 3 × 3 matrix A and nonzero real number A, we have rank(\A) = rank(A). (b) If A and B are any 2 × 2 matrices such that AB matrices C. 0, then ACB O for all 2 x 2 || (c) It is possible for a linear transformation f : R² → R² that is not the identity to be both a rotation and…

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13. Let : M₂ (R) → R defined by ( oll c C d)) = c = a + d. Recall that the operation on M₂ (R) is the usual addition of matrices. Determine, with proof, if is a group homomorphism. If it is, find its Kernel.