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4.12.E: Exercises for Section 4.9

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    Exercise \(\PageIndex{1}\)

    Apply the Gram-Schmidt process to the following set of vectors in \( \mathbb{R}^2 \) to create an orthonormal basis: \[ \vec{v}_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \quad \vec{v}_2 = \begin{bmatrix} 1 \\ -1 \end{bmatrix} \nonumber\]

    Answer

    \[ \left\{ \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ -1 \end{bmatrix} \right\} \nonumber \]

    Exercise \(\PageIndex{2}\)

    Find an orthonormal basis for the span of each of the following sets of vectors.

    1. \(\left[\begin{array}{r}3\\-4\\0\end{array}\right],\:\left[\begin{array}{r}7\\-1\\0\end{array}\right],\:\left[\begin{array}{r}1\\7\\1\end{array}\right]\)
    2. \(\left[\begin{array}{r}3\\0\\-4\end{array}\right],\:\left[\begin{array}{r}11\\0\\2\end{array}\right],\:\left[\begin{array}{r}1\\1\\7\end{array}\right]\)
    3. \(\left[\begin{array}{r}3\\0\\-4\end{array}\right],\:\left[\begin{array}{r}5\\0\\10\end{array}\right],\:\left[\begin{array}{r}-7\\1\\1\end{array}\right]\)
    Answer
    1. \(\left[\begin{array}{c}\frac{3}{5} \\ -\frac{4}{5} \\ 0\end{array}\right],\:\left[\begin{array}{c}\frac{4}{5}\\ \frac{3}{5} \\ 0\end{array}\right],\:\left[\begin{array}{c}0\\0\\1\end{array}\right]\)
    2. \(\left[\begin{array}{c}\frac{3}{5}\\ 0\\ -\frac{4}{5}\end{array}\right],\:\left[\begin{array}{c}\frac{4}{5} \\ 0\\ \frac{3}{5}\end{array}\right],\:\left[\begin{array}{c}0\\1\\0\end{array}\right]\)
    3. \(\left[\begin{array}{c}\frac{3}{5}\\0\\-\frac{4}{5}\end{array}\right],\:\left[\begin{array}{c}\frac{4}{5}\\0\\ \frac{3}{5}\end{array}\right],\:\left[\begin{array}{c}0\\1\\0\end{array}\right]\)
    Exercise \(\PageIndex{3}\)

    Using the Gram Schmidt process find an orthonormal basis for the following span: \[\text{span}\left\{\left[\begin{array}{r}1\\2\\1\end{array}\right],\:\left[\begin{array}{r}2\\-1\\3\end{array}\right],\:\left[\begin{array}{r}1\\0\\0\end{array}\right]\right\}\nonumber\]

    Answer

    A solution is \[\left[\begin{array}{c}\frac{1}{6}\sqrt{6} \\ \frac{1}{3}\sqrt{6} \\ \frac{1}{6}\sqrt{6}\end{array}\right],\:\left[\begin{array}{c}\frac{3}{10}\sqrt{2} \\ -\frac{2}{5}\sqrt{2} \\ \frac{1}{2}\sqrt{2}\end{array}\right],\:\left[\begin{array}{c}\frac{7}{15}\sqrt{3} \\ -\frac{1}{15}\sqrt{3} \\ -\frac{1}{3}\sqrt{3}\end{array}\right]\nonumber\]

    Exercise \(\PageIndex{4}\)

    Using the Gram Schmidt process find an orthonormal basis for the following span: \[\text{span}\left\{\left[\begin{array}{r}1\\2\\1\\0\end{array}\right],\:\left[\begin{array}{r}2\\-1\\3\\1\end{array}\right],\:\left[\begin{array}{r}1\\0\\0\\1\end{array}\right]\right\}\nonumber\]

    Answer

    Then a solution is \[\left[\begin{array}{c}\frac{1}{6}\sqrt{6} \\ \frac{1}{3}\sqrt{6} \\ \frac{1}{6}\sqrt{6} \\ 0\end{array}\right],\:\left[\begin{array}{c}\frac{1}{6}\sqrt{2}\sqrt{3} \\ -\frac{2}{9}\sqrt{2}\sqrt{3} \\ \frac{5}{18}\sqrt{2}\sqrt{3} \\ \frac{1}{9}\sqrt{2}\sqrt{3}\end{array}\right],\:\left[\begin{array}{c}\frac{5}{111}\sqrt{3}\sqrt{37} \\ \frac{1}{133}\sqrt{3}\sqrt{37} \\ -\frac{17}{333}\sqrt{3}\sqrt{37} \\ \frac{22}{333}\sqrt{3}\sqrt{37}\end{array}\right]\nonumber\]

    Exercise \(\PageIndex{5}\)

    Consider the following vectors: \[ \vec{v}_1 = \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \quad \vec{v}_2 = \begin{bmatrix} a \\ b \end{bmatrix} \]

    1. Find an orthonormal basis.
    2. What restriction did you have to make for your answer to part a?
    Answer
    1. \[ \left\{ \frac{1}{\sqrt{5}} \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \frac{1}{\sqrt{a^2 + b^2 - \frac{(a + 2b)^2}{5}}} \begin{bmatrix} a - \frac{a + 2b}{5} \\ b - \frac{2(a + 2b)}{5} \end{bmatrix} \right\} \nonumber \]
    2. \( b \neq 2a\).
    Exercise \(\PageIndex{6}\)

    The set \(V=\left\{\left[\begin{array}{c}x\\y\\z\end{array}\right] :2x+3y-z=0\right\}\) is a subspace of \(\mathbb{R}^3\). Find an orthonormal basis for this subspace.

    Answer

    The subspace is of the form \[\left[\begin{array}{c}x\\y\\2x+3y\end{array}\right]\nonumber\] and a basis is \(\left[\begin{array}{c}1\\0\\2\end{array}\right],\:\left[\begin{array}{c}0\\1\\3\end{array}\right]\). Therefore, an orthonormal basis is \[\left[\begin{array}{c}\frac{1}{5}\sqrt{5} \\ 0\\ \frac{2}{5}\sqrt{5}\end{array}\right],\:\left[\begin{array}{c}-\frac{3}{35}\sqrt{5}\sqrt{14} \\ \frac{1}{14}\sqrt{5}\sqrt{14} \\ \frac{3}{70}\sqrt{5}\sqrt{14}\end{array}\right]\nonumber\]

    Exercise \(\PageIndex{7}\)

    Use the Gram Schmidt process to transform the given basis for a subspace in \(\mathbb{R}^n\) into an orthogonal basis for the subspace.

    a. \(\left[\begin{array}{r}3\\0\\-3\end{array}\right],\:\left[\begin{array}{r}1\\3\\0\end{array}\right]\)

    b. \(\left[\begin{array}{r}1\\1\\0\end{array}\right],\:\left[\begin{array}{r}0\\1\\1\end{array}\right],\:\left[\begin{array}{r}1\\0\\1\end{array}\right]\)

    This page titled 4.12.E: Exercises for Section 4.9 is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Doli Bambhania, Fatemeh Yarahmadi, and Bill Wilson.