Give dim H and the E-coordinate map.
Advanced Engineering Mathematics
10th Edition
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Transcribed Image Text:a) Give \(\dim H\) and the \(\mathcal{E}\)-coordinate map.
![Given the subspace
\[ H = \left\{ \begin{bmatrix} a & 0 \\ b & c \end{bmatrix} \ \middle| \ a, b, c \in \mathbb{R} \right\}. \]
Thus, \( H \) is the space of lower triangular \( 2 \times 2 \)-matrices. The set
\[ \mathcal{E} = \left\{ \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \right\} \]
is a basis of \( H \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e470dde-d985-471c-b0b3-08e4bb7380ab%2Fae7fa5e4-3f5f-4728-ba15-a8913e928e88%2Fvji956g_processed.png&w=3840&q=75)
Transcribed Image Text:Given the subspace
\[ H = \left\{ \begin{bmatrix} a & 0 \\ b & c \end{bmatrix} \ \middle| \ a, b, c \in \mathbb{R} \right\}. \]
Thus, \( H \) is the space of lower triangular \( 2 \times 2 \)-matrices. The set
\[ \mathcal{E} = \left\{ \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \right\} \]
is a basis of \( H \).
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