91影视

Consider the matrix$B=\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]$such that$\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]A=A\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]$where$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$.

Substitute the value $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ for A in the equation $\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]A=A\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]$as follows.

$$\begin{gathered} \left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]A=A\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right] \\ \left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right] \\ \left[\begin{array}{cc}a& b\\ 2c& 2d\end{array}\right]=\left[\begin{array}{cc}a& 2b\\ c& 2d\end{array}\right] \end{gathered}$$

Compare the equation on both side as follows.

$\begin{array}{l}a=a\\ b=2b\\ 2c=c\\ 2d=2d\end{array}$

Simplify the equation$b=2b$as follows.

$\begin{array}{l}b=2b\\ b=0\end{array}$

Simplify the equation$2c=c$as follows.

$\begin{array}{l}2c=c\\ c=0\end{array}$

Simplify the equation$2d=2d$as follows.

$\begin{array}{l}2d=2d\\ d=d\end{array}$

Substitute the values 0 for c and 0 for b in the equation $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$as follows.

Therefore, any matrix in the form $A=\left[\begin{array}{cc}a& 0\\ 0& d\end{array}\right]$satisfy the equation $\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]A=A\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]$.

Simplify the equation$A=\left[\begin{array}{cc}a& 0\\ 0& d\end{array}\right]$ as follows.

$$\begin{gathered} A=\left[\begin{array}{cc}a& 0\\ 0& d\end{array}\right] \\ A=\left[\begin{array}{cc}a& 0\\ 0& d\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& d\end{array}\right] \\ A=a\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]+b\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right] \end{gathered}$$

where $\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$and $\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$are linear independent.

Therefore, the matrixS is spanned by$Span\left\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\right\}$.

Hence, the dimension of Sis 2 and spanned by $Span\left\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\right\}$.