91Ó°ÊÓ

Consider the transformation as follows.

$TinT\left(M\right)=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]M-M\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]from{R}^{(2×2)}to{R}^{(2×2)}.$

from to .

Now, check if $ker\left(t\right)=\left\{0\right\}$as follows.

$ker\left(t\right)=\left\{A∈R^{2×2}\overline{)T\left(A\right)=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]}\right\}$

Consider a matrix A as follows.

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

The next equation as follows.

localid="1659415942986" $$\begin{gathered} T\left(A\right)=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]-\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \left[\begin{array}{cc}a+0& 2a+b\\ a+d& 2c+d\end{array}\right]-\left[\begin{array}{cc}a+2c& b+2d\\ 0+c& 0+d\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \left[\begin{array}{cc}a-a-2c& 2a+b-b-2d\\ c-c& 2c+d-d\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \end{gathered}$$

Simplify further as follows.

$\left[\begin{array}{cc}-2c& 2a-2d\\ 0& 2c\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

Equating the corresponding entries as follows.

$$\begin{gathered} 2c=0 \\ 2a=2d=0 \\ 0=0 \\ 2c=0 \end{gathered}$$

From, the values are as follows.

$$\begin{gathered} c=0 \\ a=d \end{gathered}$$, with is arbitrary.

Here, does not found.

Therefore, conclude that is arbitrary.

Thus, the kernel of T with a set A as follows.

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

A set A has two dimensional with basis as follows.

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]and\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$

Hence, the nullity for the transformation is 2.