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Vector space of 2 x 2 matrices with real entries, under the usual tion and scalar multiplication. Let N C M be the set of all invertible 2 x 2 matrices. Is N a vector subspace of Mor not? Justify. 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). 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. Define a function Tr: M→ R (the trace) by the sum of the diagonal entries. Check that Tr is a linear transformation. What is the dimension of ker(Tr)?

Vector space of 2 x 2 matrices with real entries, under the usual tion and scalar multiplication. Let N C M be the set of all invertible 2 x 2 matrices. Is N a vector subspace of Mor not? Justify. 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). 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. Define a function Tr: M→ R (the trace) by the sum of the diagonal entries. Check that Tr is a linear transformation. What is the dimension of ker(Tr)?

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) Let M be the vector space of 2 x 2 matrices with real entries, under the usual addition and scalar multiplication. F (a) Let N C M be the set of all invertible 2 x 2 matrices. Is N a vector subspace of M or not? Justify. HABERTA J (b) Let J: MM 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 your basis of ker(J) as a subset. (d) Define a function Tr: MR (the trace) by the sum of the diagonal entrics. Check that Tr is a linear transformation. What is the dimension of ker(Tr)?
Transcribed Image Text:(Vector spaces) Let M be the vector space of 2 x 2 matrices with real entries, under the usual addition and scalar multiplication. F (a) Let N C M be the set of all invertible 2 x 2 matrices. Is N a vector subspace of M or not? Justify. HABERTA J (b) Let J: MM 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 your basis of ker(J) as a subset. (d) Define a function Tr: MR (the trace) by the sum of the diagonal entrics. Check that Tr is a linear transformation. What is the dimension of ker(Tr)?
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