1.3 Exercises

20 points · 4 extra credit

Vector Equations

1. 1. (2 pts) Let $\mathbf{u} = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix} 1 \\ 0 \\ -2 \end{bmatrix}$. Compute $3\mathbf{u} - 2\mathbf{v}$.

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2. 2. (3 pts) Determine whether $\mathbf{b} = \begin{bmatrix} 5 \\ 3 \end{bmatrix}$ is a linear combination of $\mathbf{a}_1 = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $\mathbf{a}_2 = \begin{bmatrix} 3 \\ -1 \end{bmatrix}$. If so, give the weights.

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3. 3. (3 pts) Determine whether $\mathbf{b} = \begin{bmatrix} 3 \\ 2 \\ -1 \end{bmatrix}$ lies in $\operatorname{Span}\{\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\}$, where

   $\mathbf{a}_1 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix},\quad \mathbf{a}_2 = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix},\quad \mathbf{a}_3 = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}.$

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4. 4. (3 pts) Describe geometrically, in $\mathbb{R}^3$:

   (a) $\operatorname{Span}\left\{\begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix}\right\}$

   (b) $\operatorname{Span}\left\{\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\right\}$

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5. 5. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) The vector equation $x_1\mathbf{a}_1 + x_2\mathbf{a}_2 + x_3\mathbf{a}_3 = \mathbf{b}$ has the same solution set as the linear system whose augmented matrix is $[\,\mathbf{a}_1 \; \mathbf{a}_2 \; \mathbf{a}_3 \mid \mathbf{b}\,]$.

   (b) The span of a single nonzero vector in $\mathbb{R}^3$ is a plane.

   (c) Asking whether $\mathbf{b} \in \operatorname{Span}\{\mathbf{a}_1, \dots, \mathbf{a}_p\}$ is the same as asking whether the system $[\,\mathbf{a}_1 \; \cdots \; \mathbf{a}_p \mid \mathbf{b}\,]$ is consistent.

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6. 6. (3 pts) Let $\mathbf{a}_1 = \begin{bmatrix} 1 \\ -2 \end{bmatrix}$ and $\mathbf{a}_2 = \begin{bmatrix} -3 \\ 6 \end{bmatrix}$. Describe exactly which vectors $\mathbf{b} = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}$ belong to $\operatorname{Span}\{\mathbf{a}_1, \mathbf{a}_2\}$, and give the defining equation of that set.

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7. 7. (3 pts) Explain why $\mathbf{0}$ belongs to $\operatorname{Span}\{\mathbf{v}_1, \dots, \mathbf{v}_p\}$ for any vectors $\mathbf{v}_1, \dots, \mathbf{v}_p$. Then explain, for two vectors $\mathbf{v}_1, \mathbf{v}_2$ in $\mathbb{R}^3$, the geometric possibilities for $\operatorname{Span}\{\mathbf{v}_1, \mathbf{v}_2\}$.

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8. 8. (4 pts ••) Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\mathbb{R}^n$. Prove that $\operatorname{Span}\{\mathbf{u}, \mathbf{v}\} = \operatorname{Span}\{\mathbf{u}, \mathbf{u} + \mathbf{v}\}$.

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