Question

Question: In Exercises 1 through 20, find the redundant column vectors of the given matrix A “by inspection.” Then find a basis of the image of A and a basis of the kernel of A.

16.$\left[\begin{array}{ccccc}1& -2& 0& -1& 0\\ 0& 0& 1& 5& 0\\ 0& 0& 0& 0& 1\end{array}\right]$

Short answer

The redundant column vectors of matrix A $\overrightarrow{v_2}\mathrm{and}\overrightarrow{{\mathrm{v}}_4}$.

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

The basis of the kernel of A =$$\begin{gathered} \left\{\left[2 \\ 1 \\ 0 \\ 0 \\ 0\right],\left[1 \\ 0 \\ -5 \\ 1 \\ 0\right]\right\} \end{gathered}$$

Step-by-step solution

Finding the redundant vectors

Let$$\begin{gathered} \overrightarrow{v_1}=\left[1 \\ 0 \\ 0\right],\overrightarrow{v_2}=\left[-2 \\ 0 \\ 0\right],\overrightarrow{v_3}=\left[0 \\ 1 \\ 0\right],\overrightarrow{v_4}=\left[-1 \\ 5 \\ 0\right],\overrightarrow{v_5}=\left[0 \\ 0 \\ 1\right] \end{gathered}$$

Here we have $\overrightarrow{v_2}=2\overrightarrow{v_1}\mathrm{and}\overrightarrow{{\mathrm{v}}_4}=-\overrightarrow{{\mathrm{v}}_3}+5\overrightarrow{{\mathrm{v}}_3}$

$\Rightarrow \overrightarrow{v_2}$are $\overrightarrow{v_4}$redundant vectors and$\overrightarrow{v_1},\overrightarrow{v_3}\mathrm{and}\overrightarrow{{\mathrm{v}}_5}$ are non-redundant vectors.

  Finding the basis of the image of A

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

Since column vectors $\overrightarrow{v_1},\overrightarrow{v_3}\mathrm{and}\overrightarrow{{\mathrm{v}}_5}$non-redundant vectors of matrix A, thus the basis of the image A =$$\begin{gathered} \left\{\left[1 \\ 0 \\ 0\right],\left[0 \\ 1 \\ 0\right],\left[0 \\ 0 \\ 1\right]\right\} \end{gathered}$$ i.e. .

Finding the basis of the kernel of A

$\overrightarrow{v_2}\mathrm{and}\overrightarrow{{\mathrm{v}}_4}$Since the vectors are redundant vectors such that

$$\begin{gathered} \overrightarrow{v_2}=2\overrightarrow{v_1}\mathrm{and}\overrightarrow{{\mathrm{v}}_4}=-\overrightarrow{{\mathrm{v}}_1}+5\overrightarrow{{\mathrm{v}}_3} \\ \Rightarrow 2\overrightarrow{{\mathrm{v}}_1}+\overrightarrow{{\mathrm{v}}_2}+0\mathrm{and}\overrightarrow{{\mathrm{v}}_1}-5\overrightarrow{{\mathrm{v}}_3}+\overrightarrow{{\mathrm{v}}_4}=0 \end{gathered}$$

Thus the vectors in kernel of A is $$\begin{gathered} \overrightarrow{w_1}=\left[2 \\ 1 \\ 0 \\ 0 \\ 0\right],\overrightarrow{w_2}=\left[1 \\ 0 \\ -5 \\ 1 \\ 0\right] \end{gathered}$$.

Final Answer

The redundant column vectors of matrix A are $\overrightarrow{v_2}\mathrm{and}\overrightarrow{{\mathrm{v}}_4}$.

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

The basis of the kernel of A =$$\begin{gathered} \left\{\left[2 \\ 1 \\ 0 \\ 0 \\ 0\right],\left[1 \\ 0 \\ -5 \\ 1 \\ 0\right]\right\} \end{gathered}$$