I posed this question (sort of) here: A 2x2 matrix $M$ exists. Suppose $M^3=0$ show that (I want proof) $M^2=0$
Suppose I have a linear transformation over a field $K$
$T:K^3\rightarrow K^3$
Suppose rank(T)=2 (it maps $K^3$ to a plane in $K^3$)
Now suppose I apply T to the result.
The second T maps a plane inside $K^3$ to a plane, so there exists a transformation $P:K^2\rightarrow K^3$ where a vector $(x,y)\in K^2$ are the coordinates of a point on the plane. P then maps this point to it's position in $K^3$
This means the transformation TT may be expressed as TP
If we let these be A and B respectively:
$A:K^3\rightarrow K^3$ but all we can tell is that rank(A)$\le$3
$B:K^2\rightarrow K^3$ and we can tell rank(B)$\le$2
(I have proved (and it was not hard) that for $L:K^m\rightarrow K^n$ rank(K)$\le$m, this is unsurprising and not difficult.)
We can express the dimensions "used" (for lack of a better term) in the target space as "rank", but how do we express the number of dimensions we "use" for the domain of the transform? Most of the time it will be the dimensions of the domain, of course.
I am looking for a notation to help me write this, please correct my terminology if it is wrong.
If a transform has a rank less than the dimensions of it's domain, if we compose that with itself, the rank of the composition will be less than the rank of the transform (which is less than the dimensions of the domain)
Having a notation for this would make statements like the above very easy to say, I'm not sure how to express them ("Dimensions we use" feels wrong) even in English right now.
Addendum
I assert above that there exists a transformation from a 2 dimensional space to a plane within a 3 dimensional space, that represents a transformation from 3 dimensional space to the same plane.
This seems logical "intuitively", and I have no doubt of it the matrix would have columns of any 2 linearly independent vectors that represent that plane (these could be directions or positions, because the plane would have to go through 0, thus a position vector could be used as a direction.
Can I prove this? (How can I prove this?)
