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(Vector spaces) Let M be the vector space of 2 x 2 matrices with real entries, under the usual addition and scalar multiplication. (a) Let N C M be the set of all invertible 2 × 2 matrices. Is N a vector subspace of M or not? Justify. (b) Let J: M → M be the function given by J(A) = A - AT. Check that J is a lincar transformation. Find a basis for the subspace S ker (J). # (c) Extend your basis from (b) to a basis of M. In other words, find a basis of M that contains CI

(Vector spaces) Let M be the vector space of 2 x 2 matrices with real entries, under the usual addition and scalar multiplication. (a) Let N C M be the set of all invertible 2 × 2 matrices. Is N a vector subspace of M or not? Justify. (b) Let J: M → M be the function given by J(A) = A - AT. Check that J is a lincar transformation. Find a basis for the subspace S ker (J). # (c) Extend your basis from (b) to a basis of M. In other words, find a basis of M that contains CI

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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**Vector Spaces Problem** Let \( M \) be the vector space of \( 2 \times 2 \) matrices with real entries, under the usual addition and scalar multiplication. (a) Let \( N \subseteq M \) be the set of all invertible \( 2 \times 2 \) matrices. Is \( N \) a vector subspace of \( M \) or not? Justify. (b) Let \( J : M \rightarrow M \) be the function given by \( J(A) = A - A^T \). Check that \( J \) is a linear transformation. Find a basis for the subspace \( S = \ker(J) \). (c) Extend your basis from (b) to a basis of \( M \). In other words, find a basis of \( M \) that contains your basis of \(\ker(J)\) as a subset.
Transcribed Image Text:**Vector Spaces Problem** Let \( M \) be the vector space of \( 2 \times 2 \) matrices with real entries, under the usual addition and scalar multiplication. (a) Let \( N \subseteq M \) be the set of all invertible \( 2 \times 2 \) matrices. Is \( N \) a vector subspace of \( M \) or not? Justify. (b) Let \( J : M \rightarrow M \) be the function given by \( J(A) = A - A^T \). Check that \( J \) is a linear transformation. Find a basis for the subspace \( S = \ker(J) \). (c) Extend your basis from (b) to a basis of \( M \). In other words, find a basis of \( M \) that contains your basis of \(\ker(J)\) as a subset.
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