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Consider the transformation as follows.

$\mathrm{T}\mathrm{in}\left(\mathrm{M}\right)=\mathrm{M}\left[\begin{array}{cc}1& 2\\ 3& 6\end{array}\right]\mathrm{from}{\mathrm{R}}^{2脳2}\mathrm{to}{\mathrm{R}}^{2脳2}$

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

data-custom-editor="chemistry" $\ker \left(\mathrm{T}\right)=\left\{\mathrm{A}鈭�{\mathrm{R}}^{2脳2}|\mathrm{T}\left(\mathrm{A}\right)=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\right\}$

Consider a matrix A as follows.

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

The next equation as follows.

role="math" localid="1659409862664" $$\begin{gathered} T\left(A\right)=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}1& 2\\ 3& 6\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \left[\begin{array}{c}a+3b2a+6b\\ c+3d2c+6d\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \end{gathered}$$

The kernel of T consist of all matrix which is transformed into zero matrix.

That is all matrices as follows.

$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$such that as follows.

$$\begin{gathered} T\left(\left[\begin{array}{c}a+3b2a+6b\\ c+3d2c+6d\end{array}\right]\right)=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ a+3b=0 \\ 鈬�a=-3b \end{gathered}$$

Simplify further as follows.

$$\begin{gathered} c+3d=0 \\ 鈬�c=-3d \end{gathered}$$

From the above calculation the values are as follows.

$$\begin{gathered} c=-3d \\ ,withb,darearbitary \\ a=-3b \end{gathered}$$

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

$$\begin{gathered} A=\left\{\left[\begin{array}{cc}-3b& b\\ -3d& d\end{array}\right]|b,darbitary\right\} \\ =\left\{b\left[\begin{array}{cc}-3& 1\\ 0& 0\end{array}\right]+b\left[\begin{array}{cc}0& 0\\ -3& 1\end{array}\right]|b,darbitrary\right\} \end{gathered}$$

A set A has two dimensional with basis as follows.

$\left[\begin{array}{cc}-3& 1\\ 0& 0\end{array}\right]and\left[\begin{array}{cc}0& 0\\ -3& 1\end{array}\right]$

Hence, the nullity for the transformation is 2.