Question 4.1.15: Find a basis of R2×2, the space of all 2 × 2 matrices, and t......
Find a basis of ℝ^{2×2}, the space of all 2 × 2 matrices, and thus determine the dimension of ℝ^{2×2}.
We can write any 2 × 2 matrix \begin{bmatrix}a&b\\c&d\end{bmatrix} as
\begin{bmatrix}a&b\\c&d\end{bmatrix}=a\begin{bmatrix}1&0\\0&0\end{bmatrix}+b\begin{bmatrix}0&1\\0&0\end{bmatrix}+c\begin{bmatrix}0&0\\1&0\end{bmatrix}+d\begin{bmatrix}0&0\\0&1\end{bmatrix}.
This shows that matrices
\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}
span ℝ^{2×2}. The four matrices are also linearly independent: None of them is a linear combination of the others, since each has a 1 in a position where the three others have a 0. This shows that
𝔅=\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)
is a basis (called the standard basis of ℝ^{2×2}), so that dim(ℝ^{2×2}) = 4.
The 𝔅-coordinate transformation L_𝔅 is represented in the following diagram:
A=\begin{bmatrix}a&b\\c&d\end{bmatrix} in ℝ^{2×2}\xrightarrow{L_𝔅}[A]_𝔅=\begin{bmatrix}a\\b\\c\\d\end{bmatrix} in ℝ^{4}.