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 $\displaystyle R^4$, 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.