1. Let W = {A E M„ (R): Tr(A) = 0}. Prove that W is a subspace of M„(R). Hint: Complete the following steps: i. Prove the following Lemma (a "Lemma" is a statement that will be used in the proof of another result): Lemma If A E M„(IR) and k E R, then Tr(kA) = k Tr(A). Verify that 0 E W, and so W + Ø. Use the result of Problem 1 in Problem Set 1 to show that if A, B E W, then A + B E W. Use the Lemma from step i to show that if A E W and k E R, then kA E W. Use Theorem 4.2.1 on p. 192 to conclude that W is a subspace of Mn (R). ii. ii. iv. V.
1. Let W = {A E M„ (R): Tr(A) = 0}. Prove that W is a subspace of M„(R). Hint: Complete the following steps: i. Prove the following Lemma (a "Lemma" is a statement that will be used in the proof of another result): Lemma If A E M„(IR) and k E R, then Tr(kA) = k Tr(A). Verify that 0 E W, and so W + Ø. Use the result of Problem 1 in Problem Set 1 to show that if A, B E W, then A + B E W. Use the Lemma from step i to show that if A E W and k E R, then kA E W. Use Theorem 4.2.1 on p. 192 to conclude that W is a subspace of Mn (R). ii. ii. iv. V.
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. Let W = {A E M, (R): Tr(A) = 0}. Prove that W is a subspace of M„(R).
Hint: Complete the following steps:
i.
Prove the following Lemma (a "Lemma" is a statement that will be used in the proof of
another result):
Lemma If A E M,(R) and k e R, then Tr(kA) = k Tr(A).
Verify that 0 E W, and so W # Ø.
Use the result of Problem 1 in Problem Set 1 to show that if A, B E W, then A + BE W.
Use the Lemma from step i to show that if A E W and k ER, then kA E W.
Use Theorem 4.2.1 on p. 192 to conclude that W is a subspace of M, (R).
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ii.
iii.
iv.
v.
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