QuestionLet $\mathbf{v}_{1}=\left[\begin{array}{r}1 \\ 0 \\ -1\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{l}4 \\ 1 \\ 5\end{array}\right], \mathbf{v}_{3}=\left[\begin{array}{r}7 \\ 2 \\ 11\end{array}\right]$, and $\mathbf{w}=\left[\begin{array}{l}5 \\ 1 \\ 4\end{array}\right]$. a. Is $\mathbf{w}$ in $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ ? How many vectors are in $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ ? b. How many vectors are in $\operatorname{Span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ ? c. Is $\mathbf{w}$ in the subspace spanned by $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ ? Why? a. Is $\mathbf{w}$ in $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ ? A. Vector $\mathbf{w}$ is not in $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ because it is not a linear combination of $\mathbf{v}_{1}, \mathbf{v}_{2}$, and $\mathbf{v}_{3}$. B. Vector $\mathbf{w}$ is in $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ because the subspace generated by $\mathbf{v}_{1}, \mathbf{v}_{2}$, and $\mathbf{v}_{3}$ is $\mathbb{R}^{3}$. C. Vector $\mathbf{w}$ is not in $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ because it is not $\mathbf{v}_{1}, \mathbf{v}_{2}$, or $\mathbf{v}_{3}$. D. Vector $\mathbf{w}$ is in $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ because it is a linear combination of $\mathbf{v}_{1}, \mathbf{v}_{2}$, and $\mathbf{v}_{3}$.
How many vectors are in $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ ? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The number of vectors in $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is $\square$ . B. There are infinitely many vectors in $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$. b. How many vectors are in $\operatorname{Span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ ? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The number of vectors in $\operatorname{Span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is $\square$ $\square$. B. There are infinitely many vectors in Span $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$.