5. Prove the following:(a) P(T) is the identity matrix if and only if T =(b ) P(πρ πι) = Ρ(π) P (πι) for each πι, T Ε S.(c) P(7)-1 = P(r-1) = P(7)*.We might say that P(7) is orthogonal since P(T)' = P(7)-!.%3D

Permutations: A permutation π of m elements is the function π:1,2,...,m→1,2,...,m given by the 2×m…

Answered: 5. Prove the following: (a) P(T) is the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Find the standard matrices A and A' for T = T2 ∘ T1 and T' = T1 ∘ T2.T1: R2 → R2, T1(x, y) = (x − 5y, 2x + 3y)T2: R2 → R2, T2(x, y) = (0, x)
13. Let - (23) M = and T be the linear operator on V defined by T(A) = MA. Find the matrix representation of T relative to the above usual basic of V.
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4. We consider 3 ×3 matrices over R. An orthogonal matrix Q such that det(Q) = 1 is called a rotation matrix. Let 1 < p
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4. Let E: P3 R2 be given by E (p(x)) = [P(2)] (a) Find a matrix that induces the transformation E. TH (b) Find a polynomial p(z) where E(p(x)) = [], or explain why this isn't possible. (c) Find a polynomial q(2) where E(q(z)) = [], or explain why this isn't possible. 8P (d) Find a nonzero polynomial r(z) where E(r(2)) = [] or explain why this isn't possible. (e) Describe the kernel of E. What is the dimension of the kernel? Find a basis for the kernel. (f) Describe the image of E. What is the dimension of the image? Find a basis for the image. (g) Is the transformation E one to one? Is it onto? Is it an isomorphism? Explain.
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5. Prove the following: (a) P(T) is the identity matrix if and only if a = idſn] - (b) P(T2 0 T1) = P(12)P(T1) for each T1, T2 E Sn. (c) P(7)-1 = P(r-1) = P(x)*. We might say that P(7) is orthogonal since P(7)' = P(7)-!.