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4.5.E: Exercise for Section 4.4

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    197431

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    Exercise \(\PageIndex{1}\)

    Here are some vectors. \[\left[\begin{array}{r}1\\1\\-2\end{array}\right],\:\left[\begin{array}{r}1\\2\\-2\end{array}\right],\:\left[\begin{array}{r}2\\7\\-4\end{array}\right],\:\left[\begin{array}{r}5\\7\\-10\end{array}\right],\:\left[\begin{array}{r}12\\17\\-24\end{array}\right]\nonumber\] Describe the span of these vectors as the span of as few vectors as possible.

    Answer

    The given vectors span the same set as: \[ \operatorname{span} \left\{ \begin{bmatrix} 1 \\ 1 \\ -2 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ -2 \end{bmatrix} \right\}. \nonumber\]

    Exercise \(\PageIndex{2}\)

    Here are some vectors. \[\left[\begin{array}{r}1\\2\\-2\end{array}\right],\:\left[\begin{array}{r}12\\29\\-24\end{array}\right],\:\left[\begin{array}{r}1\\3\\-2\end{array}\right],\:\left[\begin{array}{r}2\\9\\-4\end{array}\right],\:\left[\begin{array}{r}5\\12\\-10\end{array}\right].\nonumber\] Describe the span of these vectors as the span of as few vectors as possible.

    Exercise \(\PageIndex{3}\)

    Here are some vectors. \[\left[\begin{array}{r}1\\2\\-2\end{array}\right],\:\left[\begin{array}{r}1\\3\\-2\end{array}\right],\:\left[\begin{array}{r}1\\-2\\-2\end{array}\right],\:\left[\begin{array}{r}-1\\0\\2\end{array}\right],\:\left[\begin{array}{r}1\\3\\-1\end{array}\right]\nonumber\] Describe the span of these vectors as the span of as few vectors as possible.

    Exercise \(\PageIndex{4}\)

    Here are some vectors. \[\left[\begin{array}{r}1\\1\\-2\end{array}\right],\:\left[\begin{array}{r}1\\2\\-2\end{array}\right],\:\left[\begin{array}{r}1\\-3\\-2\end{array}\right],\:\left[\begin{array}{r}-1\\1\\2\end{array}\right]\nonumber\] Now here is another vector: \[\left[\begin{array}{r}1\\2\\-1\end{array}\right]\nonumber\] Is this vector in the span of the first four vectors? If it is, exhibit a linear combination of the first four vectors which equals this vector, using as few vectors as possible in the linear combination.

    Answer

    The vector \[ \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix} \] is in the span of the given four vectors. A minimal linear combination expressing this vector is: \[ \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix} = \frac{5}{3} \begin{bmatrix} 1 \\ 1 \\ -2 \end{bmatrix} - \frac{2}{3} \begin{bmatrix} 1 \\ 2 \\ -2 \end{bmatrix}. \nonumber\]

    Exercise \(\PageIndex{5}\)

    Here are some vectors. \[\left[\begin{array}{r}1\\1\\-2\end{array}\right],\:\left[\begin{array}{r}1\\2\\-2\end{array}\right],\:\left[\begin{array}{r}1\\-3\\-2\end{array}\right],\:\left[\begin{array}{r}-1\\1\\2\end{array}\right]\nonumber\] Now here is another vector: \[\left[\begin{array}{r}2\\-3\\-4\end{array}\right]\nonumber\] Is this vector in the span of the first four vectors? If it is, exhibit a linear combination of the first four vectors which equals this vector, using as few vectors as possible in the linear combination.

    Exercise \(\PageIndex{6}\)

    Here are some vectors. \[\left[\begin{array}{r}1\\1\\-2\end{array}\right],\:\left[\begin{array}{r}1\\2\\-2\end{array}\right],\:\left[\begin{array}{r}1\\-3\\-2\end{array}\right],\:\left[\begin{array}{r}1\\2\\-1\end{array}\right]\nonumber\] Now here is another vector: \[\left[\begin{array}{r}1\\9\\1\end{array}\right]\nonumber\] Is this vector in the span of the first four vectors? If it is, exhibit a linear combination of the first four vectors which equals this vector, using as few vectors as possible in the linear combination.

    Exercise \(\PageIndex{7}\)

    Here are some vectors, \[\left[\begin{array}{r}1\\-1\\-2\end{array}\right],\:\left[\begin{array}{r}1\\0\\-2\end{array}\right],\:\left[\begin{array}{r}1\\-5\\-2\end{array}\right],\:\left[\begin{array}{r}-1\\5\\2\end{array}\right]\nonumber\] Now here is another vector: \[\left[\begin{array}{r}1\\1\\-1\end{array}\right]\nonumber\] Is this vector in the span of the first four vectors? If it is, exhibit a linear combination of the first four vectors which equals this vector, using as few vectors as possible in the linear combination.

    Exercise \(\PageIndex{8}\)

    Here are some vectors. \[\left[\begin{array}{r}1\\-1\\-2\end{array}\right],\:\left[\begin{array}{r}1\\0\\-2\end{array}\right],\:\left[\begin{array}{r}1\\-5\\-2\end{array}\right],\:\left[\begin{array}{r}-1\\5\\2\end{array}\right]\nonumber\] Now here is another vector: \[\left[\begin{array}{r}1\\1\\-1\end{array}\right]\nonumber\] Is this vector in the span of the first four vectors? If it is, exhibit a linear combination of the first four vectors which equals this vector, using as few vectors as possible in the linear combination.

    Exercise \(\PageIndex{9}\)

    Here are some vectors. \[\left[\begin{array}{r}1\\0\\-2\end{array}\right],\:\left[\begin{array}{r}1\\1\\-2\end{array}\right],\:\left[\begin{array}{r}2\\-2\\-3\end{array}\right],\:\left[\begin{array}{r}-1\\4\\2\end{array}\right]\nonumber\] Now here is another vector: \[\left[\begin{array}{r}-1\\-4\\2\end{array}\right]\nonumber\] Is this vector in the span of the first four vectors? If it is, exhibit a linear combination of the first four vectors which equals this vector, using as few vectors as possible in the linear combination.

    Exercise \(\PageIndex{10}\)

    Suppose \(\{\vec{x}_1,\cdots ,\vec{x}_k\}\) is a set of vectors from \(\mathbb{R}^n\). Show that \(\vec{0}\) is in \(\text{span}\{\vec{x}_1,\cdots ,\vec{x}_k\}\).

    Answer

    \(\sum\limits_{i=1}^k 0\vec{x}_k=\vec{0}\)

    Exercise \(\PageIndex{11}\)

    Prove that the span of any set of vectors contains the zero vector.

    Exercise \(\PageIndex{12}\)

    Consider the vectors \( \vec v_1 = \langle 2, 3, -1 \rangle\) and \( \vec v_2 = \langle 0, 1, 2 \rangle\).

    1. Describe \( \text{span}(\vec v_1, \vec v_2)\) geometrically.
    2. Express \( \text{span}(\vec v_1, \vec v_2)\) in standard form for the type of object you concluded it is in part a.

    Answer

    1. Geometric Interpretation: The span of the two given vectors forms a plane through the origin in \(\mathbb{R}^3\).
    2. Standard Form of the Plane: The equation of the plane is: \[ 7x - 4y + 2z = 0. \nonumber\].
    Exercise \(\PageIndex{13}\)

    Let \( S = \{ \vec{v}_1, \vec{v}_2, \dots, \vec{v}_n \} \) be a set of vectors that spans a vector space \( V \). Suppose one of the vectors, say \( \vec{v}_i \), is a linear combination of the other vectors in \( S \). Prove that the set \( S' = S \setminus \{\vec{v}_i\} \) still spans \( V \), i.e., \[ \text{span}(S) = \text{span}(S'). \nonumber\]

    Note: \( S \setminus \{\vec{v}_i\} \) means the set \(S\) with the vector \(\vec{v}_i\}\) removed.

    Exercise \(\PageIndex{14}\)

    Prove that if a set of vectors contains the zero vector, removing the zero vector does not change the span of the set

    This page titled 4.5.E: Exercise for Section 4.4 is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Doli Bambhania, Fatemeh Yarahmadi, and Bill Wilson.