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47. The set of all skew-symmetric n x n matrices is a subspace W of Mnxn(F) (see Exercise 28 of Section 1.3). Find a basis for W. What is the dimension of W? 28. A matrix M is called skew-symmetric if Mt = -M. Clearly, a skew- symmetric matrix is square. Let F be a field. Prove that the set W1 of all skew-symmetric n x n matrices with entries from F is a subspace of Mnxn(F). Now assume that F is not of characteristic 2 (see Ap- pendix C), and let W2 be the subspace of Mnxn(F) consisting of all symmetric n x n matrices. Prove that Mnxn(F) = W1 e W2.

47. The set of all skew-symmetric n x n matrices is a subspace W of Mnxn(F) (see Exercise 28 of Section 1.3). Find a basis for W. What is the dimension of W? 28. A matrix M is called skew-symmetric if Mt = -M. Clearly, a skew- symmetric matrix is square. Let F be a field. Prove that the set W1 of all skew-symmetric n x n matrices with entries from F is a subspace of Mnxn(F). Now assume that F is not of characteristic 2 (see Ap- pendix C), and let W2 be the subspace of Mnxn(F) consisting of all symmetric n x n matrices. Prove that Mnxn(F) = W1 e W2.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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47. The set of all skew-symmetric n x n matrices is a subspace W of Mnxn(F) (see Exercise 28 of Section 1.3). Find a basis for W. What is the dimension of W?
Transcribed Image Text:47. The set of all skew-symmetric n x n matrices is a subspace W of Mnxn(F) (see Exercise 28 of Section 1.3). Find a basis for W. What is the dimension of W?
28. A matrix M is called skew-symmetric if Mt = -M. Clearly, a skew- symmetric matrix is square. Let F be a field. Prove that the set W1 of all skew-symmetric n x n matrices with entries from F is a subspace of Mnxn(F). Now assume that F is not of characteristic 2 (see Ap- pendix C), and let W2 be the subspace of Mnxn(F) consisting of all symmetric n x n matrices. Prove that Mnxn(F) = W1 e W2.
Transcribed Image Text:28. A matrix M is called skew-symmetric if Mt = -M. Clearly, a skew- symmetric matrix is square. Let F be a field. Prove that the set W1 of all skew-symmetric n x n matrices with entries from F is a subspace of Mnxn(F). Now assume that F is not of characteristic 2 (see Ap- pendix C), and let W2 be the subspace of Mnxn(F) consisting of all symmetric n x n matrices. Prove that Mnxn(F) = W1 e W2.
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