Originally Posted by

**AllanCuz** Hey Team,

I'm not sure where to start on part a, but part b I think I have a decent start.

$\displaystyle Rank(AB) = dim(R[AB])$

We know that A maps from Fn to Fm, and B maps from Fp to Fn. So we define the left multiplication of these guys as such,

$\displaystyle L_A: F^N \to F^M $

$\displaystyle L_B: F^P \to F^N$

So to find the range of AB we can let Z be some arbitrary vector in $\displaystyle F^P$ and see for what values is it good for

$\displaystyle dim(R[AB])=dim(L_A L_B (Z))$

We can note that $\displaystyle L_B(Z) $ is a subset of $\displaystyle F^N$ so the following inequality results,

$\displaystyle dim(L_A L_B (Z)) \le dim(L_A (F^N)) = dim(A(F^N)) = Rank(A) = N $

We can further note that if $\displaystyle L_B $ is a surjective map then its range is actually equal to $\displaystyle F^N$ turning the inequality into equality.

Was the above process correct for b, and if so, are there any hints for a?

Thanks!