Basis of a matrix.

[Dodi]

Hello!

I have got question about how to determine a two different basis for this matrix.

a11	a12	a13
0	a22	a23
0	0	a33

For my point of view is that for the solution of a first basis can have all real numbers like this
a11*

3	2	7
0	-1	8
0	0	4

+a12*

6	1	5
0	2	4
0	7	12 12

+...

Is my soultion correct?

For example I know that the standard basis for a matrix

a11	a12	a13
a21	a22	a23
a31	a32	a33

can be consisted of

a11*

1	0	0
0	0	0
0	0	0

+a12*

0	1	0
0	0	0
0	0	0

+...

Thank you for your answer.

 

[renegade05]

I'm not too sure what your question is.

A basis of a matrix is the rows or columns that are linearly independent.

That is,

\(\displaystyle c1[\vec{v1}]+c2[\vec{v2}]+...+cn[\vec{vn}] = \vec{0}\) only has the trivial solution.

If you want to determine two different basis for the matrix A, you could find the basis for the sub-spaces row(A) and col(A).

Take your matrix and put it in RREF is the first step.

You can also do a check, rank(A) = dim(row(A)) = dim(col(A)).

And for a m x n matrix, nullity(A) + rank(A) = n