Can I show that the nullspace of a matrix is a basis by finding the nullity?
For example, given matrix A, I can find the nullspace of the matrix, which is the span of a set of vectors. I can also find the nullity of A (the dimension of the nullspace of A) by reducing A to Reduced Echelon Form and counting the number of free variables.
If the nullity of A is equal to the number of vectors that span the nullspace of A, can I say that those vectors spanning the nullspace is a basis of the nullspace?
EDIT* comment from @Niing
Yes. For $n$ dimensional vector space $V$, any set of $n$ vectors that span the set is a basis and it's rather easy to prove.
Hint: Take a basis of $V$ and use replacement theorem.