Thread: Proof for matrix

ok this was confusing me so any help is appreicated.

suppose A is a n by n matrix. show that the rank of A is n only if the AX = 0 has just the trivial solution.

Originally Posted by buckaroobill

ok this was confusing me so any help is appreicated.

suppose A is a n by n matrix. show that the rank of A is n only if the AX = 0 has just the trivial solution.

Meaning A has n linearly independent coloums vectors.
Thus, they form a basis. Thus, a unique representation.
Since expressing the zero coloum is possible trivially it is the only solution for it must be unique.

What do you mean by "linearly independent"?

Originally Posted by buckaroobill

What do you mean by "linearly independent"?

The coloums of the matrix are linearly independent.
Because Rank(A)=n.