In this question let W := Mat(n, R) denote the vector space of real n x n matrices with n ≥ 2, and let ACW and TCW denote the subsets of antisymmetric and upper triangular matrices respectively (defined in your notes). For A, B E W set F(A, B) = Tr(AB), H(A, B) = Tr(A(BT)). (a) Show that A and T are subspaces of W and that W = A&T. (b) State whether the following statement is True or False, giving a justification or counterexample for your answer: Both F and H are bilinear forms on W. (c) State whether the following statement is True or False, giving a justification or counterexample for your answer: Of F and H, only H gives an inner product on W.
In this question let W := Mat(n, R) denote the vector space of real n x n matrices with n ≥ 2, and let ACW and TCW denote the subsets of antisymmetric and upper triangular matrices respectively (defined in your notes). For A, B E W set F(A, B) = Tr(AB), H(A, B) = Tr(A(BT)). (a) Show that A and T are subspaces of W and that W = A&T. (b) State whether the following statement is True or False, giving a justification or counterexample for your answer: Both F and H are bilinear forms on W. (c) State whether the following statement is True or False, giving a justification or counterexample for your answer: Of F and H, only H gives an inner product on W.
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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Transcribed Image Text:(1) In this question let W := : Mat(n, R) denote the vector space of real n × n matrices with n ≥ 2, and let AC W and
TCW denote the subsets of antisymmetric and upper triangular matrices respectively (defined in your notes).
For A, B E W set
F(A, B) = Tr(AB), H(A, B) = Tr(A(BT)).
(a) Show that A and T are subspaces of W and that W = AⓇT.
(b) State whether the following statement is True or False, giving a justification or counterexample for your answer:
Both F and H are bilinear forms on W.
(c) State whether the following statement is True or False, giving a justification or counterexample for your answer:
Of F and H, only H gives an inner product on W.
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