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Let ${\stackrel{鈬赌}{v}}_1=\left[\begin{array}{c}0\\ 0\end{array}\right]{\stackrel{鈬赌}{v}}_2=\left[\begin{array}{c}1\\ 0\end{array}\right],{\stackrel{鈬赌}{v}}_3=\left[\begin{array}{c}2\\ 0\end{array}\right],{\stackrel{鈬赌}{v}}_4=\left[\begin{array}{c}0\\ 1\end{array}\right],{\stackrel{鈬赌}{v}}_5=\left[\begin{array}{c}3\\ 4\end{array}\right]$

Here we have ${\stackrel{鈬赌}{v}}_1=0.{\stackrel{鈬赌}{v}}_2,{\stackrel{鈬赌}{v}}_3=2.{\stackrel{鈬赌}{v}}_2\mathrm{and}{\stackrel{鈬赌}{\mathrm{v}}}_5,=3{\stackrel{鈬赌}{\mathrm{v}}}_2+4{\stackrel{鈬赌}{\mathrm{v}}}_4$

$鈬�{\stackrel{鈬赌}{v}}_2$are${\stackrel{鈬赌}{v}}_4$ non-redundant vectors androle="math" localid="1659424640461" ${\stackrel{鈬赌}{v}}_1,{\stackrel{鈬赌}{v}}_3\mathrm{and}{\stackrel{鈬赌}{\mathrm{v}}}_5$ are redundant vectors.

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

Since column vectors ${\stackrel{鈬赌}{v}}_2\mathrm{and}{\stackrel{鈬赌}{\mathrm{v}}}_4$are non-redundant vectors of matrix A, thus the basis of the image A =$\left\{{\stackrel{鈬赌}{v}}_2,{\stackrel{鈬赌}{\mathrm{v}}}_4\right\}$i.e. $\left\{\left[\begin{array}{c}1\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 1\end{array}\right]\right\}$

Since the vectors ${\stackrel{鈬赌}{v}}_1,{\stackrel{鈬赌}{v}}_3\mathrm{and}{\stackrel{鈬赌}{\mathrm{v}}}_5$are redundant vectors such that

role="math" localid="1659425809735" $$\begin{gathered} {\stackrel{鈬赌}{\mathrm{v}}}_1=0.{\stackrel{鈬赌}{\mathrm{v}}}_2,{\stackrel{鈬赌}{\mathrm{v}}}_3=2.{\stackrel{鈬赌}{\mathrm{v}}}_2\mathrm{and}{\stackrel{鈬赌}{\mathrm{v}}}_5=3{\stackrel{鈬赌}{\mathrm{v}}}_2+4{\stackrel{鈬赌}{\mathrm{v}}}_4 \\ 鈬�{\stackrel{鈬赌}{\mathrm{v}}}_1-0.{\stackrel{鈬赌}{\mathrm{v}}}_2=0,2{\stackrel{鈬赌}{\mathrm{v}}}_2-{\stackrel{鈬赌}{\mathrm{v}}}_3=0\mathrm{and}3{\stackrel{鈬赌}{\mathrm{v}}}_2+3{\stackrel{鈬赌}{\mathrm{v}}}_2+4{\stackrel{鈬赌}{\mathrm{v}}}_4-{\stackrel{鈬赌}{\mathrm{v}}}_5=0 \\ \mathrm{Thus}\mathrm{the}\mathrm{vectors}\mathrm{in}\mathrm{kernel}\mathrm{of}\mathrm{A}\mathrm{is}{\stackrel{鈬赌}{\mathrm{w}}}_1=\left[\begin{array}{c}1\\ 0\\ 0\\ 0\\ 0\end{array}\right],{\stackrel{鈬赌}{\mathrm{w}}}_2=\left[\begin{array}{c}0\\ 2\\ -1\\ 0\\ 0\end{array}\right],{\stackrel{鈬赌}{\mathrm{w}}}_3=\left[\begin{array}{c}0\\ 3\\ 0\\ 4\\ -1\end{array}\right]. \end{gathered}$$

The redundant column vectors of matrix A are ${\stackrel{鈬赌}{v}}_1,{\stackrel{鈬赌}{v}}_3\mathrm{and}{\stackrel{鈬赌}{v}}_5$

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

The basis of the kernel of A =$\left\{\left[\begin{array}{c}1\\ 0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 2\\ -1\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 3\\ 0\\ 4\\ -1\end{array}\right]\right\}$