Question

In Exercises 1through 20, find the redundant column vectors of the given matrix $\mathit{A}$“by inspection.” Then find a basis of the image ofrole="math" localid="1664275567164" $\mathit{A}$and a basis of the kernel of$\mathit{A}$.

$\left(\begin{array}{cc}1& -3\\ 2& -6\\ 3& -9\end{array}\right)$

Short answer

The relation between the vectors is, $3{\overrightarrow{v}}_1+{\overrightarrow{v}}_2=0$. The basis of the image is, $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$and the basis of the kernel is, $\left[\begin{array}{c}3\\ 1\\ 0\end{array}\right]$.

Step-by-step solution

Consider the matrix

The matrix is,

$\left(\begin{array}{cc}1& -3\\ 2& -6\\ 3& -9\end{array}\right)$

Check for the image and kernel of the matrix

The vectors are ${\overrightarrow{v}}_1=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$and ${\overrightarrow{v}}_2=\left[\begin{array}{c}-3\\ -6\\ -9\end{array}\right]$.

Find the rref of the given matrix.

$$\begin{gathered} rref\left(\begin{array}{cc}1& -3\\ 2& -6\\ 3& -9\end{array}\right)=\left(\begin{array}{cc}3& -9\\ 0& 0\\ 0& 0\end{array}\right) \\ =\left(\begin{array}{cc}1& -3\\ 0& 0\\ 0& 0\end{array}\right) \end{gathered}$$

The relation can be defined as,

role="math" localid="1664276015909" $$\begin{gathered} 3{\overrightarrow{v}}_1+{\overrightarrow{v}}_2=0 \\ \left[\begin{array}{c}{\overrightarrow{v}}_1\\ {\overrightarrow{v}}_2\\ {\overrightarrow{v}}_3\end{array}\right]=\left[\begin{array}{c}3\\ 1\\ 0\end{array}\right] \end{gathered}$$

The image of the matrix is, $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$.

The kernel of the matrix is, $\left[\begin{array}{c}3\\ 1\\ 0\end{array}\right]$.

Final answer

The relation between the vectors is, $3{\overrightarrow{v}}_1+{\overrightarrow{v}}_2=0$. The basis of the image is, $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$ and the basis of the kernel is, $\left[\begin{array}{c}3\\ 1\\ 0\end{array}\right]$ .