# Does the proof of rank-nullity theorem from Lang's “Linear Algebra” involves axiom of choice? [duplicate]

Disclaimer: I never had (yet c:) a rigorous exposure to set theory (independence proofs, and similar stuff...).

I was wondering if, in the following proof of the rank-nullity theorem form Lang's "Linear Algebra" book (page 61, theorem 3.2 in the third edition), the author makes use of the axiom of choice:

Let $V$, $W$ be vector spaces and let $L:V \to W$ be a linear map. Then we have $\dim V = \dim \ker L + \dim \text{im}\, L$.

Proof: Let $n$ be the dimension of $V$, $q$ the dimension of the kernel of $L$, and $s$ the dimension of the image of $L$. Then, assuming $s > 0$, let $\{w_1,\dots, w_s\}$ be a basis of $\text{im}\, L$; let $v_1,\dots,v_s$ be $s$ elements of $V$ such that $L(v_i) = w_i$ for $i = 1,\dots, s$. [...]

Here, the inverse image $L^{-1}(\{w_i\})$ need not to be unique, but the author claims that he can arbitrarily take $s$ elements of $V$ such that $L(v_i)=w_i$; this can be carried out with a choice function $\varphi: \{L^{-1}(\{w_i\}): i = 1,\dots, s\} \to V$: we have our $\{v_1,\dots, v_s\}$ by considering the image of $\varphi$.

Is the axiom of choice (or any weaker equivalent) required in this case?

## marked as duplicate by Asaf Karagila♦ axiom-of-choice StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jul 9 '18 at 23:42

Not at all. One can prove by induction on $n$ that for any finite $n,$ we can make $n$ choices. Choice principles come into play when we need to make infinitely-many choices, with no clear way to specify how said choices should/can be made.