(a) (b) (c) Let W be the subpace of M2x2 (R) b W = { [20 + $ 30+d]:akedER} a, b, c, Find a basis for W. You do not need to prove that it is a basis. W is isomorphic to Rn for some n. Put the value of this n in this box: Prove that W is isomorphic to R", where n is your value from part (b).
(a) (b) (c) Let W be the subpace of M2x2 (R) b W = { [20 + $ 30+d]:akedER} a, b, c, Find a basis for W. You do not need to prove that it is a basis. W is isomorphic to Rn for some n. Put the value of this n in this box: Prove that W is isomorphic to R", where n is your value from part (b).
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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Question
![Let \( W \) be the subspace of \( M_{2 \times 2}(\mathbb{R}) \)
\[
W = \left\{ \begin{bmatrix} 2a + b & 3c + d \\ -d & 0 \end{bmatrix} : a, b, c, d \in \mathbb{R} \right\}.
\]
(a) Find a basis for \( W \). You do not need to prove that it is a basis.
(b) \( W \) is isomorphic to \( \mathbb{R}^n \) for some \( n \). Put the value of this \( n \) in this box: \(\Box\)
(c) Prove that \( W \) is isomorphic to \( \mathbb{R}^n \), where \( n \) is your value from part (b).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb04829d0-4645-426e-bf1a-7ada40b0786f%2F729f94fc-6b40-4bd6-8485-30ba78681c71%2Fd875vhl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let \( W \) be the subspace of \( M_{2 \times 2}(\mathbb{R}) \)
\[
W = \left\{ \begin{bmatrix} 2a + b & 3c + d \\ -d & 0 \end{bmatrix} : a, b, c, d \in \mathbb{R} \right\}.
\]
(a) Find a basis for \( W \). You do not need to prove that it is a basis.
(b) \( W \) is isomorphic to \( \mathbb{R}^n \) for some \( n \). Put the value of this \( n \) in this box: \(\Box\)
(c) Prove that \( W \) is isomorphic to \( \mathbb{R}^n \), where \( n \) is your value from part (b).
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