# Basic question of linear algebra

Gold Member
I know the rank nullity theorem.
But can i say that the vectors in the column space of A and the vectors in the nullspace of A are linearly independent?
Thanks and this is not a hw.

Gold Member
i think the nilpotent matrix may say its not true, any thought?

no, you don't know that vectors Tv will belong in your nullspace or not (in general , but you might have an invariant transformation, or a projection). You know that v is linearly independent from your nullspace vectors if Tv is not zero

Well, I don't know if this is what you refer to, but the relation between A

and A^T seen as maps is that the nullspace of A^T is the orthogonal complement

of Im(A) (the image set of A, or, if A represents a map L:V-->W: {Ax: x in V} )

and the image of A^T is the orthogonal complement of N(A), the nullspace of A.

Gold Member
Well, I don't know if this is what you refer to, but the relation between A

and A^T seen as maps is that the nullspace of A^T is the orthogonal complement

of Im(A) (the image set of A, or, if A represents a map L:V-->W: {Ax: x in V} )

and the image of A^T is the orthogonal complement of N(A), the nullspace of A.

Well what i mean is that are the vectors in R(A) and N(A) linearly independent.

Deveno