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Answer the following questions. (a) Recall that M2,2 (R) is the set of all 2 × 2 matrices and Ei,j is a matrix where the entry ai = 1 and any other entry in Eij is equal to zero. Consider a 2 x 2 invertible matrix A. Show that M2,2 (R) = span({AE1,1, AE1.2, AE2,1, AE2,2}). (b) Consider W = {p(x) = P₁(R)|p(x) = p(-x) for very x = R}. Show that W = span({1, x², x¹}).

Answer the following questions. (a) Recall that M2,2 (R) is the set of all 2 × 2 matrices and Ei,j is a matrix where the entry ai = 1 and any other entry in Eij is equal to zero. Consider a 2 x 2 invertible matrix A. Show that M2,2 (R) = span({AE1,1, AE1.2, AE2,1, AE2,2}). (b) Consider W = {p(x) = P₁(R)|p(x) = p(-x) for very x = R}. Show that W = span({1, x², x¹}).

Advanced Engineering Mathematics
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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Answer the following questions. (a) Recall that M2,2(R) is the set of all 2 x 2 matrices and Eij is a matrix where the entry aij = 1 and any other entry in Eij is equal to zero. Consider a 2 x 2 invertible matrix A. Show that M2,2(R) = span({AE1,1, AE1,2, AE2,1, AE2,2}). (b) Consider W = {p(x) E P5(R)\p(x) = p(-x) for very r € R}. Show that W = span({1,x²,x*}).
Transcribed Image Text:Answer the following questions. (a) Recall that M2,2(R) is the set of all 2 x 2 matrices and Eij is a matrix where the entry aij = 1 and any other entry in Eij is equal to zero. Consider a 2 x 2 invertible matrix A. Show that M2,2(R) = span({AE1,1, AE1,2, AE2,1, AE2,2}). (b) Consider W = {p(x) E P5(R)\p(x) = p(-x) for very r € R}. Show that W = span({1,x²,x*}).
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