Answer
$\mathscr{B}$={$\begin{bmatrix} 1\\0\\1\\\end{bmatrix}$, $\begin{bmatrix} 0\\1\\0\\\end{bmatrix}$}
The dimension of the basis is 2.
Work Step by Step
The set of vectors that have equal first and last entries are of the form
$\begin{bmatrix} s\\t\\s\\\end{bmatrix}$, which can also be expressed by: $\mathit{s}$ $\begin{bmatrix} 1\\0\\1\\\end{bmatrix}$ + $\mathit{t}$ $\begin{bmatrix} 0\\1\\0\\\end{bmatrix}$.
Then since the vectors $\begin{bmatrix} 1\\0\\1\\\end{bmatrix}$ and $\begin{bmatrix} 0\\1\\0\\\end{bmatrix}$ are linearly independent, they also make up a basis.
Finally, since the basis contains two vectors, the dimension of the basis is 2.
