91影视

Let${\stackrel{鈫�}{v}}_1=\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right],{\stackrel{鈫�}{v}}_2=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right],{\stackrel{鈫�}{v}}_3=\left[\begin{array}{c}5\\ 0\\ 0\\ 0\end{array}\right]{\stackrel{鈫�}{v}}_4=\left[\begin{array}{c}3\\ 1\\ 0\\ 0\end{array}\right]{\stackrel{鈫�}{v}}_5=\left[\begin{array}{c}-3\\ 3\\ 0\\ 0\end{array}\right]$

Here we have ${\stackrel{鈫�}{\mathrm{v}}}_2=0.,{\stackrel{鈫�}{\mathrm{v}}}_1,{\stackrel{鈫�}{\mathrm{v}}}_3=5{\stackrel{鈫�}{\mathrm{v}}}_1\mathrm{and}{\stackrel{鈫�}{\mathrm{v}}}_4=4{\stackrel{鈫�}{\mathrm{v}}}_1+\frac{1}{3}{\stackrel{鈫�}{\mathrm{v}}}_5$.

$鈬�{\stackrel{鈫�}{\mathrm{v}}}_2,{\stackrel{鈫�}{\mathrm{v}}}_3\mathrm{and}{\stackrel{鈫�}{\mathrm{v}}}_4$are redundant vectors and ${\stackrel{鈫�}{\mathrm{v}}}_1\mathrm{and}{\stackrel{鈫�}{\mathrm{v}}}_5$are non-redundant vectors.

The non-redundant column vectors of A form the basis of the image of A.

Since column vectors ${\stackrel{鈫�}{\mathrm{v}}}_1\mathrm{and}{\stackrel{鈫�}{\mathrm{v}}}_5$are non-redundant vectors of matrix A, thus the basis of the image A = $\left\{{\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_5\right\}i.e\left\{\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}-3\\ 3\\ 0\\ 0\end{array}\right]\right\}$.

Since the vectors ${\stackrel{鈫�}{\mathrm{v}}}_2,{\stackrel{鈫�}{\mathrm{v}}}_3\mathrm{and}{\stackrel{鈫�}{\mathrm{v}}}_4$are redundant vectors such that

$$\begin{gathered} {\stackrel{鈫�}{v}}_2=0.{\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_3=5{\stackrel{鈫�}{v}}_{1}and{\stackrel{鈫�}{v}}_4=4{\stackrel{鈫�}{v}}_1+\frac{1}{3}{\stackrel{鈫�}{v}}_5 \\ 鈬�0.{\stackrel{鈫�}{v}}_1-{\stackrel{鈫�}{v}}_2=0,5{\stackrel{鈫�}{v}}_1-{\stackrel{鈫�}{v}}_3=0and4{\stackrel{鈫�}{v}}_1-{\stackrel{鈫�}{v}}_4+\frac{1}{3}{\stackrel{鈫�}{V}}_5=0 \end{gathered}$$

Thus the vectors in the kernel of A are ${\stackrel{鈫�}{w}}_1=\left[\begin{array}{c}0\\ -1\\ 0\\ 0\\ 0\end{array}\right],{\stackrel{鈫�}{w}}_2=\left[\begin{array}{c}5\\ 0\\ -1\\ 0\\ 0\end{array}\right],{\stackrel{鈫�}{w}}_3=\begin{array}{c}\left[\begin{array}{c}4\\ 0\\ 0\\ -1\\ \frac{1}{3}\end{array}\right]\end{array}$.

The redundant column vectors of matrix A are ${\stackrel{鈫�}{\mathrm{v}}}_2,{\stackrel{鈫�}{\mathrm{v}}}_3\mathrm{and}{\stackrel{鈫�}{\mathrm{v}}}_4$.

The basis of the image of A = $A=\left\{\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}-3\\ 3\\ 0\\ 0\end{array}\right]\right\}$.

The basis of the kernel of A = $\left\{\left[\begin{array}{c}0\\ -1\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}5\\ 0\\ -1\\ 0\\ 0\end{array}\right],\begin{array}{c}\left[\begin{array}{c}4\\ 0\\ 0\\ -1\\ \frac{1}{3}\end{array}\right]\end{array}\right\}$.