91Ó°ÊÓ

Consider the matrix $B=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$such that$\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]A=A\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$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 $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]A=A\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$as follows.

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

Compare the equation both side as follows.

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

Simplify the equation$a+c=a$as follows.

$$\begin{gathered} a+c=a \\ c=0 \end{gathered}$$

Simplify the equation $b+d=a+b$as follows.

$$\begin{gathered} b+d=a+b \\ d=a \end{gathered}$$

Simplify the equation$d=c+d$as follows.

$\begin{array}{l}d=c+d\\ c=0\end{array}$

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

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

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

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

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

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

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

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