91影视

For a line to be an image of a matrix $A$it has to be spanned by column vectors of $A$, and since it is spanned by a vector already we'll use it as a column vector

$A=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$

For a line to be a kernel of matrix$B$ the equality must be correct

$B=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=0$

Sincewe have no other conditions we can choose any vector that gives 0 in dot product with

$\stackrel{鈫�}{\mathrm{v}}=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$

We choose vector

$\stackrel{鈫�}{w}=\left[\begin{array}{c}1\\ 1\\ 鈭�2\end{array}\right]$

But we can construct another vector that gives 0 in dot product with $\stackrel{鈫�}{w}$so our matrix is not complete, we know that kernel of must form a plane.

So we find another vector $\stackrel{鈫�}{k}$ that is non collinear with $\stackrel{鈫�}{w}$ and it's dot product with $\stackrel{鈫�}{v}$is 0 let's say

$\stackrel{鈫�}{k}=\left[\begin{array}{c}1\\ 鈭�1\\ 0\end{array}\right]$

Now wehave two non collinear vectors so they span a plane and now we construct matrix$B$

$B=\left[\begin{array}{ccc}1& 1& 鈭�2\\ 1& 鈭�1& 0\end{array}\right]$

Matrix$A$

$A=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$

Matrix$B$

$B=\left[\begin{array}{ccc}1& 1& 鈭�2\\ 1& 鈭�1& 0\end{array}\right]$