# Linear transformation + ranks

1. Feb 10, 2010

1. The problem statement, all variables and given/known data
Let S(U)=V and T(V)=W be linear maps where U,V, W are vector spaces over the same field K. Prove :

2. Relevant equations
a) Rank (TS) <= Rank (T)
b) Rank (TS) <= Rank (S)
c) if U=V and S is nonsingular then Rank (TS) = Rank (T)
d) if V=W and T is nonsingular then Rank (TS) = Rank (S)

3. The attempt at a solution
a) TS maps to W, so is T
b) TS maps to W, but S to V, but how do I show the ranks for (a) and (b)?
c) d) So inverse of S and T exists, and err...

U,V,W are vector space over the SAME field, does that mean they have the same number of entries, say R2, R3, etc etc

2. Feb 10, 2010

### HallsofIvy

3. Feb 10, 2010

"TS maps to W, so is T"

sorry, what I meant was T also maps to W

for (c) and (d)
what does the the nonsingularity of the matrix imply?
how does that show that the ranks are both equal?

4. Feb 10, 2010

### vintwc

I think the best way to show that the ranks are equal is by considering the surjectivity and injectivity of S and T respectively (since S and T are non-singular). From there, it should be pretty much straightforward.