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".

 

[Architeuthis Dux]

for help

I would be happy to help you with your linear algebra proofs. Let's start with the first one:

1) To prove that if A^2=0, then the columns of matrice A are vectors in solution space of the system Ax=0, we can use the definition of matrix multiplication. Since A^2=0, we know that each entry of the matrix A^2 must be equal to 0. This means that for any vector x, we have (A^2)x=0. Using the definition of matrix multiplication, we can expand this to A(Ax)=0. Since the columns of A are the coefficients of the linear combination Ax, this means that each column of A must be a solution to the system Ax=0. Therefore, the columns of A are vectors in the solution space of the system Ax=0.

To show that p(A)>=n/2, we can use the fact that the rank of a matrix is equal to the number of linearly independent columns. Since we know that the columns of A are solutions to the system Ax=0, and there are at least n/2 columns (since A is a nXn matrix), the rank of A must be at least n/2. Therefore, p(A)>=n/2.

2) To prove that if p(A^2)<p(A), then the system Ax=0 has a non-trivial solution, we can use the fact that the rank of a matrix is equal to the number of linearly independent columns. If p(A^2)<p(A), this means that there are more linearly independent columns in A than in A^2. This implies that there must be at least one column in A that is not a linear combination of the columns of A^2. This column represents a non-trivial solution to the system Ax=0.

To show that the system A^2x=0 has a solution y which is Ay≠0, we can use the same logic as above. Since p(A^2)<p(A), there must be at least one linearly independent column in A^2 that is not a linear combination of the columns of A. This means that there exists a vector y such that A^2y=0, but Ay≠0. This vector y is a solution to the system A^2x=0, and since Ay≠0, it is