Answer
$\mathscr{B}$={$\begin{bmatrix}
1 \\2\\-1\\-3
\end{bmatrix}$, $\begin{bmatrix}
-4 \\5\\0\\7
\end{bmatrix}$}
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}a- 4b-2c \\ 2a +5b-4c\\-a+2c\\-3a+7b+6c\end{bmatrix}$
= $\mathit{a}$ $\begin{bmatrix}
1 \\2\\-1\\-3
\end{bmatrix}$ + $\mathit{b}$ $\begin{bmatrix}
-4 \\5\\0\\7
\end{bmatrix}$ + $\mathit{c}$ $\begin{bmatrix}
-2 \\-4\\2\\6
\end{bmatrix}$
Every vector can be written as a linear combination of the vectors
$\begin{bmatrix}
1 \\2\\-1\\-3
\end{bmatrix}$, $\begin{bmatrix}
-4 \\5\\0\\7
\end{bmatrix}$, $\begin{bmatrix}
-2 \\-4\\2\\6
\end{bmatrix}$.
However, by inspection, it is easy to see that the set is linearly dependent and therefore cannot be a basis:
-2*$\begin{bmatrix}
1 \\2\\-1\\-3
\end{bmatrix}$= $\begin{bmatrix}
-2 \\-4\\2\\6
\end{bmatrix}$.
So we must use the Spanning Set Theorem; we'll remove the last vector from the set.
Thus, we get that the basis contains the following vectors:
$\mathscr{B}$={$\begin{bmatrix}
1 \\2\\-1\\-3
\end{bmatrix}$, $\begin{bmatrix}
-4 \\5\\0\\7
\end{bmatrix}$}
Then since the basis contains two vectors, the dimension of the basis is 2.
