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 $\displaystyle R^4$, which are $\displaystyle (1,0,0,0)^t,\ (0,1,0,0)^t,\ (0,0,1,0)^t,\ (0,0,0,1)^t$
Now:
$\displaystyle
{\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 $\displaystyle (0,0,0)^t$
and as:
$\displaystyle
{\rm{image}}(T)={\rm{span}} \{(1,0,0)^t,\ (0,0,1)^t \}
$
we can let these two vectors be the images of $\displaystyle (0,1,0,0)^t$ and $\displaystyle (0,0,0,1)^t$ respectivly.
So now we have assigned images to the unit vectors in $\displaystyle R^4$ which satisfy all the given conditions.
Then clearly we can write:
$\displaystyle
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.