Linear Algebra Proofs for A^2=0 and p(A^2)<p(A)

[Anatolyz]

Hello !
i try to solve Linear algebra question(but need be written properly as mathmatical proofs)
Having A matrice nXn:
1)proove that if A^2=0 the columns of matrice A are vectors in solution space of the system Ax=0 (x and 0 are vectors of course),and show that p(A)>=n/2
2)proove that if p(A^2)<p(A) (p in all cases here means: the rank of the vectors)
so the system Ax=o has a non trivial solution and the System A^2x=0 has solution y which is Ay≠0,,,,
I have the general clue but how write it right,math way i have big problem..
thank you very much

 

[HallsofIvy]

Hello !
i try to solve Linear algebra question(but need be written properly as mathmatical proofs)
Having A matrice nXn:
1)proove that if A^2=0 the columns of matrice A are vectors in solution space of the system Ax=0 (x and 0 are vectors of course),and show that p(A)>=n/2

What is A(1, 0, 0...)^T? A(0, 1, 0,...)^T?, etc.

2)proove that if p(A^2)<p(A) (p in all cases here means: the rank of the vectors)

vectors don't have "ranks". I presume you mean the rank of A² and A.

so the system Ax=o has a non trivial solution and the System A^2x=0 has solution y which is Ay≠0,,,,
I have the general clue but how write it right,math way i have big problem..
thank you very much

If you have a "general clue" please tell us what it is. Perhaps we can help with the mathematics notation for that. I started to give a hint but I suspect it may be just your "general clue".