Is $\begin{bmatrix}2 & 3 & 0\end{bmatrix}$ in the span of $v_2 = \begin{bmatrix} 1 & 0 \end{bmatrix}$ and $v_3 = \begin{bmatrix} 0 & 1 \end{bmatrix}$

Question

Is the vector $v_1 = \begin{bmatrix}2 & 3 & 0\end{bmatrix}$ in the span of the vectors $v_2 = \begin{bmatrix} 1 & 0 \end{bmatrix}$ and $v_3 = \begin{bmatrix} 0 & 1 \end{bmatrix}$?

I believe the answer is false because $v_1$ is in $3$-dimensional space and $v_2$ and $v_3$ are in $2$-dimensional space but I am not sure if there is something I'm missing (trick question?).

Thanks.

Answer 1

Your reasoning is right, the dimension doesn't match.

Notice that $\operatorname{span}\{ v_2, v_3\}=\mathbb{R}^2$ while $v_1 \not \in \mathbb{R}^2$.