Suppose A is an n * n matrix and A^2 = A. If an vector v is in both nulA and ColA, prove that u must be zero vector.
Let v be the vector that is in nullspace,as well as column space of A
So Av=0. and Ax=v for some vector x
Now Ax=v
Imply A(Ax)=Av
Imply A2x=Av
Imply Ax=Av=0. ( Since A2=A)
Imply Ax=0
Imply v=0 (since Ax=v)
So v is a zero vector and our result is proved.
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