3.1 #20 The questions in the picture please answer a

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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3.1 #20

The questions in the picture please answer a, b

20. One Theorem, Three Proofs: The goal of this Exercise is to prove the following
Theorem in three different ways:
Theorem: For any linear transformation T : R" → R", we must have: T(0, ) = 0m-
Note: as part of your three proofs, explain clearly why 0, appears on the left side of the
equation, but 0m appears on the right side.
Use the Additivity Property of T to provide the first proof.
Hint: compute T(0, + 0n) in two different ways.
Use the Homogeneity Property of T to provide the second proof.
How should you modify the Hint from part (a)?
а.
b.
Transcribed Image Text:20. One Theorem, Three Proofs: The goal of this Exercise is to prove the following Theorem in three different ways: Theorem: For any linear transformation T : R" → R", we must have: T(0, ) = 0m- Note: as part of your three proofs, explain clearly why 0, appears on the left side of the equation, but 0m appears on the right side. Use the Additivity Property of T to provide the first proof. Hint: compute T(0, + 0n) in two different ways. Use the Homogeneity Property of T to provide the second proof. How should you modify the Hint from part (a)? а. b.
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