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Let $N = \{v_1, v_2 \ldots, v_n\}$ be a basis for null space. Then the nullity of $N$ is $n$. By definition of null space, $c_1v_1 + c_2v_2 + \ldots + c_nv_n$ is the solution to $Ax = 0$ where $c_i$ is a scalar. Since there are $n$ scalars, there are $n$ free variables.

Does that make sense?

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    $N$ is nothing but a set. The second sentence does not make sense. – 2017-01-08
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    @ Jack, I got the second sentence from the following proof: https://s30.postimg.org/8f8fapbtt/Capture.png Can you, please, explain why $(*)$ in the link considered **the** homogeneous solution to $Ax = b?$ – 2017-01-08
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    Is your question from the link your gave? You might probably want to edit your question. – 2017-01-08
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    I think you meant the nullity of $A$ not $N$? – 2017-01-08
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    @ Jack, no my question is as asked. The link is an inspiration that I can't quite make sense of. So my follow up question would be if I parsed the argument in my link incorrectly. Also, yes about the nullity of $A.$ – 2017-01-08
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    As I said, $N$ is a *set*. It does not make sense to say "the nullity of $N$". – 2017-01-08

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The rank-nullity theorem for $A : V \to W$ gives $$ \DeclareMathOperator{nul}{nul} \DeclareMathOperator{rk}{rk} \nul A + \rk A = \dim V $$ The total number of variables of the system $A x = 0$, $x \in V$, is $\dim V = n$.

The dimension of the row space of $A$ is $\rk A$, it gives the number of linear independent constraints $$ r_i \cdot x = 0 $$ on the $n$ variables for the system $A x = 0$, where $r_i$ is the $i$-th row of $A$, thus the number of not-free variables. This leaves the number of free variables as $$ n - \rk A = \nul A $$

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    Given that the proof of the rank-nullify theorem relies on the characterization of nullify by free variables, this answer is somewhat circular. – 2017-01-08
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    @Alqatrkapa What about proof 2 here: https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem#Proofs – 2017-01-08
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    Aha, indeed that proof is independent. But that isn't the usual proof for introductory linear algebra as far as I know, so some caution should still be held. – 2017-01-08
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    I understand that concern. Tempts me to go for my old lecture notes. – 2017-01-08