a) To graph: the ker(A),ker(A)^bot and im(A^T) ​ b) To find: the relationship between im(A^T) and ker(A).

a) Given:
The image of linear transformation $T\left(\bar{x}\right)=A\left(\bar{x}\right)$ is the span of the column vectors of A.
The kernel of a linear transformation $T\left(\bar{x}\right)=A\left(\bar{x}\right)\mathfrak{o}m{\mathbb{\text{R}}}^m\to {\mathbb{\text{R}}}^n$ consists of all zeros of the transformation.
That is the solution of the equation $T\left(\bar{x}\right)=A\left(\bar{x}\right)=0.$
We denote the kernel of T by ker (A) or ker (T) and it is the solution set of the linear system $A\left(\bar{x}\right)=0.$
Graph:

Interpretation:
Consider the linear system $A\left(\overrightarrow{x}\right)=\overrightarrow{b}f\text{or}A=\left[\begin{array}{cc}1& 3\\ 2& 6\end{array}\right]\text{and}\overrightarrow{b}=\left[\begin{array}{c}10\\ 20\end{array}\right].$
$A\left(\overrightarrow{x}\right)=\overrightarrow{0}$
$\left[\begin{array}{cc}1& 3\\ 2& 6\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=0$
$x_1=-3t,x_2=t$
$t=parametr$
$\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{c}-3t\\ t\end{array}\right]=\left[\begin{array}{c}-3\\ 1\end{array}\right]t$
Thus, ker (A) is the line spanned by $\left[\begin{array}{c}-3\\ 1\end{array}\right]\in {\mathbb{\text{R}}}^2.$
Consider im (A) which is spanned by ${\overrightarrow{u}}_1=\left[\begin{array}{c}1\\ 2\end{array}\right].$ Then,
${\left(kerA\right)}^{\mathrm{\perp }}=ker\left(A\right)-proj\left(kerA\right)$
${\left(kerA\right)}^{\mathrm{\perp }}=ker\left(A\right)-({\overrightarrow{u}}_1.kerA){\overrightarrow{u}}_1$
${\left(kerA\right)}^{\mathrm{\perp }}=\left[\begin{array}{c}-3\\ 1\end{array}\right]-[\left[\begin{array}{c}1\\ 2\end{array}\right]\cdot \left[\begin{array}{c}-3\\ 1\end{array}\right]]\left[\begin{array}{c}1\\ 2\end{array}\right]$
${\left(kerA\right)}^{\mathrm{\perp }}=\left[\begin{array}{c}-3\\ 1\end{array}\right]+\left[\begin{array}{c}1\\ 2\end{array}\right]$
${\left(kerA\right)}^{\mathrm{\perp }}=\left[\begin{array}{c}-2\\ 3\end{array}\right]$
Consider $A^T=\left[\begin{array}{cc}1& 2\\ 3& 6\end{array}\right].$
Since

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