7. Let R2x2 denote the vector space of real 2 x 2 matrices. Consider the 2 x 2 matrix and let W = {A € R2X2 : AX = XA}. (a) Show that W is a subspace of R2x2. (b) Find a basis for W.

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mdasfaaruddinshareefLv10

13 Jun 2023

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Jean KeelingLv2

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4. Let S= 1 1 , and consider the subspace W = Span(S) of R. (a) Show that S is an orthogonal basis for W. (b) Let x = 2. Find projw(x). 11 (c) Let x = 12 be an arbitrary vector in R3. Find projw(x) and then find a matrix A such that projw(x) = Ax.

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8. Determine if each of the following statements is true or false. Clearly indicate your answer for each part by stating if it is true or false no justification is required). Each part is worth 2 marks for a correct answer; -1 for an incorrect answer to discourage guessing) (a) If A and B are n x n matrices, then rank(BA) rank(A). (b) If {A, B, C} is a linearly independent set of n x m matrices, then {AT, BT,CT} is a linearly independent set of m x n matrices. (C) W = {A E Rnxn A is invertible } is a subspace of Rnxn, where Rnxn denotes the vector space of real n x n matrices. (d) If xi and X2 are orthogonal vectors in R3 then it is always possible to find a unique vector X3 such that {x1, X2, X3} is an orthogonal basis for R3. (e) Let W be a subspace of R" and let x ER". Then projw(x - projw(x)) = 0.

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7. Let R2x2 denote the vector space of real 2 x 2 matrices. Consider the 2 x 2 matrix and let W = {A € R2X2 : AX = XA}. (a) Show that W is a subspace of R2x2. (b) Find a basis for W.

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