Question

Give an example of a$\mathbf{4}\mathbf{\times }\mathbf{5}$matrix A with$\mathit{d}\mathit{i}\mathit{m}\mathbf{\left(}\mathbf{k}\mathbf{e}\mathbf{r}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{\right)}\mathbf{=}\mathbf{3}$$\mathit{d}\mathit{i}\mathit{m}\mathbf{\left(}\mathbf{k}\mathbf{e}\mathbf{r}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{\right)}\mathbf{=}\mathbf{3}$.

Short answer

An example of $4\times 5$matrix such$A$ that$\dim \left(\ker \left(A\right)\right)=3$ is,

$A=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

Step-by-step solution

To mention given data and recall theorem 3.3.7

Let $A$be any$4\times 5$ matrix.

We have to find the matrix$A$ such that $\dim \left(\ker \left(A\right)\right)=3$.

Theorem 3.3.7, is stated as follows:

For any$n\times m$ matrix $A$, the equation,

$\dim \left(\ker \left(A\right)\right)+\dim \left(im\left(A\right)\right)=m$.

Here, in this case $n=4,m=5$.

To give an example of4×5 matrix with dim(ker(A))=3

By Theorem 3.3.7, we have,

$\begin{array}{l}\dim \left(\ker \left(A\right)\right)+\dim \left(im\left(A\right)\right)=m\\ \Rightarrow \dim \left(\ker \left(A\right)\right)+rank\left(A\right)=m\\ \Rightarrow \dim \left(\ker \left(A\right)\right)=m-rank\left(A\right)\\ \Rightarrow \dim \left(\ker \left(A\right)\right)=5-rank\left(A\right)\end{array}$

Considering$rank\left(A\right)=2$ , where$A$ is any$4\times 5$ matrix whose rank is 2 given by,

$A=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]$.

Hence, we get,

$\begin{array}{l}\dim \left(\ker \left(A\right)\right)=5-2\\ \Rightarrow \dim \left(\ker \left(A\right)\right)=3\end{array}$