# Universal property of generating set for vector space

Let $V$ be a vector space over $F$, and $S$ a non-empty subset of $V$. We say that $S$ generates $V$ if every $v\in V$ can be written as finite $F$-linear combination of elements of $S$.

I want to express the above definition of generating set in terms of universal property. My question is whether the following is the universal property of $S$ (generating set) and how to prove that it is equivalent to above definition? Just hint is sufficient, I will try to write complete proof.

We say that $S$ generates $V$ if the following holds: any map $\varphi$ from $S$ into any vector space $W$ extends to a linear map $\varphi_1$ from $V$ into/onto $W$.

• Do you wanna qualify that $\varphi$ be a homomorphism? – AJY Jul 10 '16 at 5:28
• no, in vector space category, morphisms should be linear maps (it was initial typo from me, sorry for inconvenience, I corrected it) – p Groups Jul 10 '16 at 5:28

The universal property you wrote (leaving out any assertion about $\varphi_1$ being injective or surjective) is equivalent to $S$ being linearly independent, not generating $V$. You want to say instead that any $\varphi$ extends to at most one linear map $\varphi_1$ (again, with no condition that $\varphi_1$ is injective or surjective). The hard direction of the proof is that if $S$ has this property, then $S$ generates $V$. To prove this direction, you need to give an example of a $\varphi$ for which $\varphi_1$ is not unique, assuming $S$ does not generate $V$. To do this, let $U$ be the subspace generated by $S$, $W=V/U$, and $\varphi=0$.
• if $\varphi_1$ is onto linear map, will it not imply that $S$ generates $V$? – p Groups Jul 10 '16 at 5:30
• I don't know what you mean by that. What choice of $\varphi:S\to W$ are you making? The basic problem with your condition is that if $S$ is not linearly independent, then for most choices of $\varphi$ there will not exist any linear extension $\varphi_1$. – Eric Wofsey Jul 10 '16 at 5:32
• OK, I will make it precise: let $S$ be a subset of $V$ satisfying following property: given any map from $S$ to a vector space $W$, it extends to a linear map from $V$ onto $W$ (i.e. $\varphi_1\colon V\rightarrow W$ is surjective linear map). Then is this (property) equivalent to say that $S$ generates $V$? – p Groups Jul 10 '16 at 5:34
• That condition is not true for any subset of any vector space. You can always choose $W$ to have larger dimension than $V$, and then there can't exist any linear surjection $V\to W$. – Eric Wofsey Jul 10 '16 at 5:35