What is the dimension of \(\mathbb{R}^{2\times 3}\), the vector space of
\(2 \times 3\) matrices with real entries?
The answer is 6.
A basis for \(\mathbb{R}^{2\times 3}\) is given by
\(\begin{bmatrix} 1 & 0 & 0\\0 & 0 & 0 \end{bmatrix}\),
\(\begin{bmatrix} 0 & 1 & 0\\0 & 0 & 0 \end{bmatrix}\),
\(\begin{bmatrix} 0 & 0 & 1\\0 & 0 & 0 \end{bmatrix}\),
\(\begin{bmatrix} 0 & 0 & 0\\1 & 0 & 0 \end{bmatrix}\),
\(\begin{bmatrix} 0 & 0 & 0\\0 & 1 & 0 \end{bmatrix}\),
\(\begin{bmatrix} 0 & 0 & 0\\0 & 0 & 1 \end{bmatrix}\).