Advanced Engineering Mathematics
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:

\[
\langle \vec{a} \vec{v}, \vec{u}_1 \rangle \vec{u}_1 + \langle \vec{v}, \vec{u}_2 \rangle \vec{u}_2.
\]
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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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