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Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![Let \( \mathcal{S} \) be a set containing the two column matrices shown:
\[
\mathcal{S} = \left\{ \left\{ \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \right\}, \left\{ \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \right\} \right\}.
\]
Let the first column be \( \vec{u}_1 \) and the second column be \( \vec{u}_2 \).
Let \( \vec{v} = \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} \).
Find the projection of \( \vec{v} \) onto \( \mathcal{S} \). First normalize \( \mathcal{S} \). Then calculate:
\[
\langle \vec{a} \vec{v}, \vec{u}_1 \rangle \vec{u}_1 + \langle \vec{v}, \vec{u}_2 \rangle \vec{u}_2.
\]](https://content.bartleby.com/qna-images/question/494356ae-2db7-448e-a147-892d80022497/05eb94ce-c59a-4b77-85af-ef26f8d382e0/8b1a6qs_thumbnail.png)
Transcribed Image Text:Let \( \mathcal{S} \) be a set containing the two column matrices shown:
\[
\mathcal{S} = \left\{ \left\{ \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \right\}, \left\{ \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \right\} \right\}.
\]
Let the first column be \( \vec{u}_1 \) and the second column be \( \vec{u}_2 \).
Let \( \vec{v} = \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} \).
Find the projection of \( \vec{v} \) onto \( \mathcal{S} \). First normalize \( \mathcal{S} \). Then calculate:
\[
\langle \vec{a} \vec{v}, \vec{u}_1 \rangle \vec{u}_1 + \langle \vec{v}, \vec{u}_2 \rangle \vec{u}_2.
\]
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