Linear Algebra writing the right way?

[Anatolyz]

Hello !
i try to solve Linear algebra 2 questions(but need them 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 clue but how write it rught,math way i have big problem..
thank you very much

 

[Unco]

Hi Anatolyz,

To begin to write an answer to the question "1) Prove that if A^2=0, then the columns of the matrix A are vectors in the solution space of the system Ax=0", begin by writing something like: "Let v be a column of A." Your task is then to show that Av = 0.

To do so, remember that we can write \(\displaystyle A^2 = AA\) as

\(\displaystyle AA = A[v_1 \, v_2\, \cdots \, v_n] = [Av_1 \, Av_2\, \cdots \, Av_n]\)

where \(\displaystyle v_i\) is the \(\displaystyle i^{\text{th}}\) column vector of A.

If you proceed from there and subequently get stuck, show us your work so we can keep you chugging along.