Question

2. Let ? = {E, E, E, E} be the standard ordered basis for M???(?), where M???(?) is the vector space of all 2 2 real matrices under the usual matrix addition and scalar multiplication. Let T: M???(?) ? M???(?) the linear operator defined by T(B) = B?, the transpose of B, for each B ? M???(?) (a) (4 pts) Use definition to compute matrix A = [T]?. Determine all values ? such that det(A ? ?I) = 0, where I is the identity 4 4 matrix. Show your work. (b) (8 pts) For each eigenvalue ? obtained in (a), find its corresponding eigenspace E? = N(A ? ?I) = {x ? ?? : (A ? ?I)x = 0} and determine a basis for E?. Show your work. (c) (3 pts) Use (b) to show that there exists a basis ? for ?? such that Q?AQ is diagonal matrix for some invertible matrix Q as in Theorem 2.23 (Section 2.5 on Page 113) and the paragraph after Example 6 in Section 5.1 on Pages 251-252.

          2. Let ? = {E, E, E, E} be the standard ordered basis for M???(?), where M???(?) is the vector space of all 2    2 real matrices under the usual matrix addition and scalar multiplication. Let T: M???(?) ? M???(?) the linear operator defined by T(B) = B?, the transpose of B, for each B ? M???(?)

(a) (4 pts) Use definition to compute matrix A = [T]?. Determine all values ? such that det(A ? ?I) = 0, where I is the identity 4    4 matrix. Show your work.

(b) (8 pts) For each eigenvalue ? obtained in (a), find its corresponding eigenspace E? = N(A ? ?I) = {x ? ?? : (A ? ?I)x = 0} and determine a basis for E?. Show your work.

(c) (3 pts) Use (b) to show that there exists a basis ? for ?? such that Q?AQ is diagonal matrix for some invertible matrix Q as in Theorem 2.23 (Section 2.5 on Page 113) and the paragraph after Example 6 in Section 5.1 on Pages 251-252.

2. Let ? = E, E, E, E be the standard ordered basis for M???(?), where M???(?) is the vector space of all 2 2 real matrices under the usual matrix addition and scalar multiplication. Let T: M???(?) ? M???(?) the linear operator defined by T(B) = B?, the transpose of B, for each B ? M???(?)

(a) (4 pts) Use definition to compute matrix A = [T]?. Determine all values ? such that det(A ? ?I) = 0, where I is the identity 4 4 matrix. Show your work.

(b) (8 pts) For each eigenvalue ? obtained in (a), find its corresponding eigenspace E? = N(A ? ?I) = x ? ?? : (A ? ?I)x = 0 and determine a basis for E?. Show your work.

(c) (3 pts) Use (b) to show that there exists a basis ? for ?? such that Q?AQ is diagonal matrix for some invertible matrix Q as in Theorem 2.23 (Section 2.5 on Page 113) and the paragraph after Example 6 in Section 5.1 on Pages 251-252.

Added by Erin M.

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