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4.2E: Exercises for Section 4.1

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    197425

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    Exercise \(\PageIndex{1}\)

    Find \(-3\left[\begin{array}{c}5\\-1\\2\\-3\end{array}\right]+5\left[\begin{array}{c}-8\\2\\-3\\6\end{array}\right]\).

    Answer

    \(\left[\begin{array}{c}-55\\13\\-21\\39\end{array}\right]\)

    Exercise \(\PageIndex{2}\)

    Find \(-7\left[\begin{array}{c}6\\0\\4\\-1\end{array}\right]+6\left[\begin{array}{c}-13\\-1\\1\\6\end{array}\right]\).

    Exercise \(\PageIndex{3}\)

    Let

    \[ \vec{a} = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}, \quad \vec{b} = \begin{bmatrix} -4 \\ 5 \\ 0 \end{bmatrix}. \nonumber \]

    Compute:

    1. \( \vec{a} + \vec{b} \)
    2. \( 3\vec{a} \)
    3. \( -2\vec{b} \)
    4. \( \vec{a} - \vec{b} \)
    Exercise \(\PageIndex{4}\)

    Let \( \vec{v} \) and \( \vec{w} \) be two vectors in \( \mathbb{R}^2 \). How is the vector sum \( \vec{v} + \vec{w} \) related to the diagonal of the parallelogram formed by \( \vec{v} \) and \( \vec{w} \)?

    Exercise \(\PageIndex{5}\)

    Let \[ \vec{v} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}. \nonumber \]

    1. What happens to the magnitude of \( \vec{v} \) when it is multiplied by \(2\)?
    2. What happens to the direction of \( \vec{v} \) when it is multiplied by \(-1\)?
    Exercise \(\PageIndex{6}\)

    Let \(\vec v_1\) and \(\vec v_2\) be vectors in \( \mathbb{R}^2 \). What geometric shape does the set \(\{a\vec v_1+b\vec v_2 \,\big{|}\, 0 \le a \le 1, 0 \le b \le 1\}\)

    Exercise \(\PageIndex{7}\)

    Decide whether \[\vec{v}=\left[\begin{array}{c}4\\4\\-3\end{array}\right]\nonumber\] is a linear combination of the vectors \[\vec{u}_{1}=\left[\begin{array}{c}3\\1\\-1\end{array}\right]\quad\text{and}\quad\vec{u}_{2}=\left[\begin{array}{c}2\\-2\\1\end{array}\right].\nonumber\]

    Answer

    \[\left[\begin{array}{c}4\\4\\-3\end{array}\right]=2\left[\begin{array}{c}3\\1\\-1\end{array}\right]-\left[\begin{array}{c}2\\-2\\1\end{array}\right]\nonumber\]

    Exercise \(\PageIndex{8}\)

    Decide whether \[\vec{v}=\left[\begin{array}{c}4\\4\\4\end{array}\right]\nonumber\] is a linear combination of the vectors \[\vec{u}_1=\left[\begin{array}{c}3\\1\\-1\end{array}\right]\quad\text{and}\quad\vec{u}_2=\left[\begin{array}{c}2\\-2\\1\end{array}\right].\nonumber\]

    Answer

    The system \[\left[\begin{array}{c}4\\4\\4\end{array}\right]=a_1\left[\begin{array}{c}3\\1\\-1\end{array}\right]+a_2\left[\begin{array}{c}2\\-2\\1\end{array}\right]\nonumber\] has no solution.

    Exercise \(\PageIndex{9}\)

    Decide whether \[\vec{v}=\left[\begin{array}{c}2\\-2\\1\end{array}\right]\nonumber\] is a linear combination of the vectors \[\vec{u}_1=\left[\begin{array}{c}1\\0\\-1\end{array}\right]\quad\text{,}\quad\vec{u}_2=\left[\begin{array}{c}5\\-4\\0\end{array}\right] \text{and}\quad\vec{u}_3=\left[\begin{array}{c}2\\-3\\1\end{array}\right].\nonumber\]

    Exercise \(\PageIndex{10}\)
    1. Prove that there exists a unique zero vector \( \vec{0} \) such that for every vector \( \vec{a} \in \mathbb{R}^n \), \[ \vec{a} + \vec{0} = \vec{a}. \nonumber\]
    2. Prove that for every vector \( \vec{a} \in \mathbb{R}^n \), there exists a unique vector \( -\vec{a} \) such that \[ \vec{a} + (-\vec{a}) = \vec{0}. \nonumber\
    Exercise \(\PageIndex{11}\)

    Let \( \vec{a} \) and \( \vec{b} \) be two vectors in \( \mathbb{R}^n \). Show geometrically that

    \[ \|\vec{a} + \vec{b} \|^2 + \|\vec{a} - \vec{b} \|^2 = 2\|\vec{a}\|^2 + 2\|\vec{b}\|^2. \nonumber\]

    Hint

    Sketch the parallelogram formed by the vectors \( \vec{a} \) and \( \vec{b} \). Notice that the two diagonal vectors of this parallelogram are given by: \[ \text{One diagonal: } \quad \vec{a} + \vec{b} \nonumber\] \[ \text{Other diagonal: } \quad \vec{a} - \vec{b} \nonumber \]

    Then use the Law of Cosines.

    This page titled 4.2E: Exercises for Section 4.1 is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Doli Bambhania, Fatemeh Yarahmadi, and Bill Wilson.