QuestionDescribe the Span $\{\mathbf{v}_{1}, \mathbf{v}_{2}\}$ for $\mathbf{v}_{1}=\begin{bmatrix}4 \\ 10 \\ -2\end{bmatrix}$ and $\mathbf{v}_{2}=\begin{bmatrix}10 \\ 25 \\ -5\end{bmatrix}$ . Choose A, B, C, or D.

Studdy Solution

STEP 1

Assumptions1. We have two vectors $\mathbf{v}_{1}=\left[\begin{array}{r}4 \\10 \\ -\end{array}\right]$ and $\mathbf{v}_{}=\left[\begin{array}{r}10 \\25 \\ -5\end{array}\right]$ in $\mathbb{R}^{3}$ . . We want to find the geometric description of Span $\left\{\mathbf{v}_{1}, \mathbf{v}_{}\right\}$ .

STEP 2

First, we need to check if the vectors $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ are linearly independent. Two vectors are linearly independent if there is no scalar $c$ such that $\mathbf{v}_{1} = c \mathbf{v}_{2}$ .

STEP 3

Let's check if there is a scalar $c$ such that $\mathbf{v}_{1} = c \mathbf{v}_{2}$ .

STEP 4

We can see that $\mathbf{v}_{2} =2. \mathbf{v}_{1}$ , which means that $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ are not linearly independent.

STEP 5

Since the vectors are not linearly independent, the Span $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$ is the set of all points on the line through $\mathbf{v}_{1}$ and $\mathbf{0}$ .
So, the correct answer is A. Span $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$ is the set of points on the line through $\mathbf{v}_{1}$ and $\mathbf{0}$ .

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