1. ## More linear Algebra

Find an example of a linear transformation Transpose: R^4--->R^3 such that
ker T= span{(1,0,0,0),(0,0,1,0)}
and
im T= span{(1,0,0),(0,0,1)}.

Fully justify example.

2. Originally Posted by natewalker205
Find an example of a linear transformation Transpose: R^4--->R^3 such that
ker T= span{(1,0,0,0),(0,0,1,0)}
and
im T= span{(1,0,0),(0,0,1)}.

Fully justify example.
The transformation is uniquely defined by the images of the unit vectors in $R^4$, which are $(1,0,0,0)^t,\ (0,1,0,0)^t,\ (0,0,1,0)^t,\ (0,0,0,1)^t$

Now:

$
{\rm{ker}}(T)={\rm{span}} \{(1,0,0,0)^t,\ (0,0,1,0)^t \}
$

tells us that the images of these two vectors are both $(0,0,0)^t$

and as:

$
{\rm{image}}(T)={\rm{span}} \{(1,0,0)^t,\ (0,0,1)^t \}
$

we can let these two vectors be the images of $(0,1,0,0)^t$ and $(0,0,0,1)^t$ respectivly.

So now we have assigned images to the unit vectors in $R^4$ which satisfy all the given conditions.

Then clearly we can write:

$
T = \left[ {\begin{array}{*{20}c}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}} \right]
$

as one such matrix.