The answer is “True”.
The span of
\(\left\{
\begin{bmatrix} 1\\0\\0\\0\end{bmatrix},
\begin{bmatrix} 0\\1\\0\\0\end{bmatrix},
\begin{bmatrix} 0\\0\\1\\0\end{bmatrix},
\begin{bmatrix} 0\\0\\0\\1\end{bmatrix}
\right\}\) with real numbers as scalars is \(\mathbb{R}^4\).