Answer
The dimension of the basis is 2.
Work Step by Step
We are given that every vector $\vec{x}$ can be expressed by
$\vec{x}$= $\begin{bmatrix}
s-2t \\
s + t\\
3t\\
\end{bmatrix}$
= $\mathscr{s}$ $\begin{bmatrix}
1 \\
1\\
0\\
\end{bmatrix}$ + $\mathscr{t}$ $\begin{bmatrix}
-2 \\
1\\
3\\
\end{bmatrix}$ .
Every vector can be written as a linear combination of the vectors
$\begin{bmatrix}
1 \\
1\\
0\\
\end{bmatrix}$ and $\begin{bmatrix}
-2 \\
1\\
3\\
\end{bmatrix}$, therefore these vectors make up a basis for the set.
$\mathscr{B}$ = {$\begin{bmatrix}
1 \\
1\\
0\\
\end{bmatrix}$ , $\begin{bmatrix}
-2 \\
1\\
3\\
\end{bmatrix}$}.
Since there are two vectors in the basis, its dimension is 2.
