How are the spanning vectors and null space of a matrix related?

Let's say we have a matrix $A$.

From what I understand, the null space of A is the set of all vectors $v$, such that $A * v = 0$.

Also from my understanding, the spanning vectors of A are all the solutions to the homogeneous system that A represents. Is the span of the spanning vectors of A the null space of A?

I do believe there is a relationship somewhere I just don't see it.

Thank you.

• I have never heard the term 'spanning vectors' used for the null space. (Then again, I haven't heard of many things...) – copper.hat Feb 5 '17 at 18:13
• Which textbook are you using? That might help us understand your question better. – Omnomnomnom Feb 5 '17 at 22:51

I quite don't get the notion of spanning vectors of A. For me a matrix is a linear transformation from $V \to W$ where V is a vector space of dimension n , W is of m( when the matrix is $n \times m$ )
Now we have a set of vectors $v_1,,v_n ;w_1,w_2..w_m$ basis for V and W. Now the null space(kernel) is spanned by some set of $v_i$'s and the image is spanned by some set of $w_i$'s The rank-nullity theorem states that $\dim(\ker A) + \dim( im(A)) = n$. Hope this helps.