Answer
The set of vectors cannot be linearly independent due to the presence of the zero vector
Case 1: The set does not form a basis.
Case 2: Since $M$ is not invertible the columns cannot span $R^3$
Work Step by Step
From the given Vectors;
$\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}0\\0\\0\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}$
We need to inspect the vectors to determine which sets in bases for $R^3$
The set of vectors cannot be linearly independent due to the presence of the zero vector
Case 1: The set does not form a basis.
We can row reduce the matrix by combining the vectors to form a Matrix $M$.
let ${R_3} = {R_3} - {R_1}$
$M = \begin{bmatrix}1&0&0\\0&0&1\\1&0&0\end{bmatrix}=\begin{bmatrix}1&0&0\\0&0&1\\0&0&0\end{bmatrix}$
From this result we can see that the Matrix has only two Pivot points.
Case 2: Since $M$ is not invertible the columns cannot span $R^3$
