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 , which are
Now:
tells us that the images of these two vectors are both
and as:
we can let these two vectors be the images of and respectivly.
So now we have assigned images to the unit vectors in which satisfy all the given conditions.
Then clearly we can write:
as one such matrix.