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Homework Statement
Show that for all nxn matricies A with real entries we have
rk(A*A) = rk(A) where A* is the transpose of A.
Homework Equations
The Attempt at a Solution
I'm working over a vector space V.
Im(A) = {A(v) | vEV}
Im(A*A) = {A*(A(v)) | vEV}
So Im(A*A) is a subset of Im(A)
So rk(A*A) =< rk(A)
Ker(A) = {A(w) = 0 | wEV}
Ker(A*A) = {A*(A(w)) = 0 | wEV}
so Ker(A*A) is a subset of Ker(A)
so dimKer(A*A) =< dimKer(A)
dimKer(A) = dimV - rk(A)
so -rk(A*A) =< -rk(A)
so rk(A*A) >= rk(A)
Combining the results yields rk(A*A) = rk(A)
Is this correct??