I'm teaching linear algebra and one of my students came to me with a different way to find the kernel and the image of a linear transformation, that doesn't use Gauss-Jordan reduction:
SOLUTION:
- The matrix $A$ is
$$ A=\left(\begin{array}{ccc} -14 & -6 & 2 \\ -10 & -4 & 2 \\ 6 & 2 & -2 \end{array}\right) $$
- The kernel of $L$ is the subspace of $\mathbb{R}^{3}$ containing the vectors $\boldsymbol{v}=x, y, z$ such that it yields $L(\boldsymbol{v})=\mathbf{0}$, that is
$$ \left(\begin{array}{ccc} -14 & -6 & 2 \\ -10 & -4 & 2 \\ 6 & 2 & -2 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\left(\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right) $$
which is an algebraic linear system having rank 2 , because
$$ \operatorname{det}\left(\begin{array}{ccc} -14 & -6 & 2 \\ -10 & -4 & 2 \\ 6 & 2 & -2 \end{array}\right)=0 $$
and the minor of order 2
$$ \left(\begin{array}{cc} -4 & 2 \\ 2 & -2 \end{array}\right) $$
highlighted in the matrix $A$
$$ A=\left(\begin{array}{ccc} -14 & -6 & 2 \\ -10 & -4 & 2 \\ 6 & 2 & -2 \end{array}\right) $$
has determinant not equal to zero. From this minor we get the system
$$ \left\{\begin{array}{l} -4 y+2 z=10 t \\ 2 y-2 z=-6 t \end{array}\right. $$
in which we have given the arbitrary value $x=t$ to the unknown $x$, that lays outside the minor highlighted in the matrix $A$. The solution is then
$$ \left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\left(\begin{array}{c} 1 \\ -2 \\ 1 \end{array}\right) t $$
that is the kernel is spanned by the basis vector $(1,-2,1)$. The image of $L$ is spanned by all those vectors whose components are contained inside the minor highlighted in the matrix $A$, that is we have the basis of the image
$$ \mathcal{B}_{I m(L)}=\left\{\left(\begin{array}{c} -3 \\ -2 \\ 1 \end{array}\right),\left(\begin{array}{c} 1 \\ 1 \\ -1 \end{array}\right)\right\}, $$
where the second and third column of $A$ have been divided by 2 .
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I know how to solve it using G-J reduction, and I understand what he does to explain why $rank(A)=2$ but after that, I don't have any idea what he's using and I didn't find in the bibliography or Google some kind of possible explanation. (about how he get the basis of kernel and image)
Can someone explain the reasoning to me?