I'm trying to understand a point made in a lecture video by Benedict Gross. The context, paraphrased, is as follows.
We have a finite-dimensional vector space $V$ and a subspace $W$ of $V$. We define the canonical quotient map $f: V \to V/W$. Let $\{w_1, \ldots, w_n\}$ be a basis for $W$. As the $w_i$ are linearly independent in $W$ and hence in $V$, we can extend them to a basis $\{w_1, \ldots, w_n, v_{n+1}, v_{n+2}, \ldots, v_m$. Now let $W' = \text{span}(v_{n+1}, v_{n+2}, \ldots, v_m)$.
Gross then says: "Because of the existence of bases, you can find a subspace which in some sense lifts this." I believe by "this" he was referring to $V/W$.
I'm having a lot of trouble understanding this comment. My understanding of lifts, based on the Wikipedia article, is as follows. Let $f: X \to Y$ and $g: Z \to Y$ be maps. It seems to define a "lift" in the context of the function $f$, rather than a set or a subspace, as a lift of $f$ to another set, say $Z$ (so the two necessary components of a lift are the function and the set it's being lifted to). The "lift" is then a map $h: X \to Z$ with $f = g \circ h$.
I'm having difficulty reconciling this with Gross's comment. I can't seem to "compose" maps per se, but thinking of this in terms of a diagram, I could define the inclusion map from $W'$ to $V$ in addition to the isomorphism from $W'$ to $V/W$, but this doesn't seem in any way "commutative."
Best Answer
Let me try and provide some intuition. Quotients appear all throughout mathematics, not just for vector spaces. Given an object $X$ (in your case a finite dimensional vector space), and a subobject $Y$ of $X$ (in your case a vector subspace), we can define a quotient $X/Y$ and a quotient map $\pi: X \to X/Y$ (in your case, a linear map). Then, informally, if we have a "thing" defined on $X/Y$, we can often "lift" it to a "thing" defined on $X$. The canonical example is that if we have a map $f: X/Y \to Z$ (in your case a linear map), we can lift it to a map $f\circ\pi: X\to Z$ by precomposing by $\pi$.
Here it seems Gross uses the word "lift" not for the lifting of a map, but for lifting the vector space $V/W$ itself. Here's how you can do so: choose a basis $\{w_1,\dots,w_n\}$ for $W$. Extend it to a basis $\{w_1,\dots,w_n,v_{n+1},\dots,v_m\}$ for $V$. Then $\{v_{n+1}+W,\dots,v_m+W\}$ is a basis for $V/W$. So we can "lift" $V/W$ to $V$ with the linear map $\iota: V/W\to V$ defined on basis vectors by $$\iota(v_i+W) =v_i.$$ So this allows to view $V/W$ as $\iota(V/W)$, a subspace of $V$ (Note that $\iota$, unlike $\pi$, is not canonical! It depends on the choice of basis extension $\{v_{n+1},\dots,v_m\}$. The subspace $\iota(V/W)$ is not canonical either!).
Keep in mind the term "lift" is informal, and can mean different things depending on the "thing" we are lifting.