例题 2.2. (2阶实矩阵)

考虑实数域上的所有 $2\times 2$ 矩阵的全体关于矩阵的加法和数乘构成的线性空间 $M_{22}\left(\mathbb{R}\right)$.

解

设

$${\mathbf{E}}_{11}=\left(\begin{array}{ll}1& 0\\ 0& 0\end{array}\right),{\mathbf{E}}_{12}=\left(\begin{array}{ll}0& 1\\ 0& 0\end{array}\right),{\mathbf{E}}_{21}=\left(\begin{array}{ll}0& 0\\ 1& 0\end{array}\right),{\mathbf{E}}_{22}=\left(\begin{array}{ll}0& 0\\ 0& 1\end{array}\right).$$

首先我们证明 ${\mathbf{E}}_{11},{\mathbf{E}}_{12},{\mathbf{E}}_{21},{\mathbf{E}}_{22}$ 线性无关. 若 $k_1{\mathbf{E}}_{11}+k_2{\mathbf{E}}_{12}+k_3{\mathbf{E}}_{21}+$ $k_4{\mathbf{E}}_{22}=0$,即有 $\left(\begin{array}{ll}{k}_1& k_2\\ k_3& k_4\end{array}\right)=\left(\begin{array}{ll}0& 0\\ 0& 0\end{array}\right)$,故得 $k_1=k_2=k_3=k_4=0$. 又对任意 $2\times 2$ 矩阵 $\mathbf{A}=\left(\begin{array}{ll}a& b\\ c& d\end{array}\right)$ 有

$$\mathbf{A}=a{\mathbf{E}}_{11}+b{\mathbf{E}}_{12}+c{\mathbf{E}}_{21}+d{\mathbf{E}}_{22}$$

所以 ${\mathbf{E}}_{ij}\left(i,j=1,2\right)$ 是 $M_{22}\left(\mathbb{R}\right)$ 的一组基, $\dim \left(M_{22}\left(\mathbb{R}\right)\right)=4$ 且 $\mathbf{A}$ 在基 ${\mathbf{E}}_{11},{\mathbf{E}}_{12}$, ${\mathbf{E}}_{21},{\mathbf{E}}_{22}$ 下的坐标为 $\left(a,b,c,d\right)$.