# Find a basis for the kernel and image of matrix A

I need to find a basis for the kernel and image of the matrix A

$$A = \left( \begin{array}{ccc} -12 & 6 \\ 4 & -2\\ -8 & 4 \end{array} \right)$$

But I am unsure how to do that.

For the kernel

$$A \left( \begin{array}{ccc} x \\ y \end{array} \right)= \left( \begin{array}{ccc} 0\\ 0\\ 0\end{array} \right)$$ , I get that x = 2y.

For the Image

$$A \left( \begin{array}{ccc} x \\ y \end{array} \right)= \left( \begin{array}{ccc} a\\ b\\ c\end{array} \right)$$ , I get c=-2b, b=-(1/3)1 OR (1/3)a+3b+c=0

How do I write those into a basis?

• There is a standard procedure to find a basis for the kernel and image of a matrix $A$. You can row-reduce $A$ to row-echelon or reduced row-echelon form. The columns that are leading in the row-echelon form tell you which columns from the original matrix $A$ you can take to get a basis for the image of $A$. For the kernel, use your row-echelon form to solve $A\mathbf{x}=\mathbf{0}$, and you will get your answer in terms of a linear combination of certain vectors: those vectors form a basis for the kernel. If you have any worked examples in your notes, those would be good to look over. – Minus One-Twelfth Mar 21 at 2:26
• I cant reduce this to reduced row echelon form because after row reduction the second and third rows become (0 0) and cant be used to reduce the first row. Plus, in this question , both basis are to be a single vector (1 vector for kernel and 1 for image) – Mohamad Moustafa Mar 21 at 2:29
• The kernel of $A$ is all linear combintations of $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ i.e. $\ker A = \mathrm{span}\,\left\{\begin{pmatrix} 1 \\ 2 \end{pmatrix}\right\}$. Since the columns of $A$ are linearly dependent, we can take the span of either column to be the image of $A$. – Brian Mar 21 at 2:39
• @Brian Thanks, what about the image basis? I need a single vector that can serve as a basis. – Mohamad Moustafa Mar 21 at 2:42
• @MohamadMoustafa What is one vector that is in the image/ column space of $A$? How does the column space relate to this one vector? – Brian Mar 21 at 2:45