Q:

Linear Algebra - Clarifying the meaning of a lemma related to linear maps

I am reading through Jim Hefferon's "Linear Algebra" and have encountered a lemma which I have had some difficulty in understanding. Through reading various other resources, I believe I currently understand the lemma but I am not complete sure. I would appreciate it if someone could confirm or correct my understanding.
The lemma is located near the end of Chapter 3, Section II.1 and reads as follows:

For vector spaces V and W, the set of linear functions from V to W is itself a vector space, a subspace of the space of all functions from V to W.

What I believe this means is that there is a set of linear functions, $G = \{G_1, G_2, ...G_n\}$ that map $V$ to $W$ such that these maps conform to the rules of vector spaces. As an example for all $G_k, G_m \in G,$ we would have closure: $G_k+G_m \in G,$ associativity under addition: $G_k+G_m = G_m+G_k$, additive identity: $G_k + G_n = G_k, G_n \in G$, etc.
Further, I believe the portion of lemma that reads:

"... subspace of the space of all functions from V to W."

states that there are some non-linear functions that map from $V$ to $W$ and that the functions we would include in $G$ (as defined in the previous paragraph) would be the subset of of these functions which are linear.
Is this a correct interpretation?

A:

Yes. You may want to check that the set is indeed a vector space just to convince yourself. 

