Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![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).](https://content.bartleby.com/qna-images/question/b04829d0-4645-426e-bf1a-7ada40b0786f/729f94fc-6b40-4bd6-8485-30ba78681c71/d875vhl_processed.jpeg)
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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