# finite and infinite, vector space, linear transformation

I have to answer these questions for homework and I don't know if I'm answering these correctly. I think most of them are correct, but a double check would be much appreciated.

a) If $S$ is a set of vectors from $V$, what is $\operatorname{span}(S)$? Discuss also the case when $S$ is an infinite set.

So for this I think $\operatorname{span}(S)$ is all possible linear combination of vectors in $S$. For infinite the set must be countable.

b) If $S$ is a set of vectors from $V$, what does it mean for $S$ to be linearly independent? Discuss also the case when $S$ is an infinite set.

So a set of vectors is L.I. if there is a non-trivial solution where at least one component is not zero. And for infinite, $S$ is L.I. if all the sub-spaces of $S$ are L.I.

c) If $V$ is a vector space, what is a sub-space $W$ of $V$?

Definition:A non-empty subset $W$ of a vector space $(V, \oplus, \odot)$ is called a subspace of $V$ if $W$ is closed under $\oplus$ and $\odot$. $V$ is ambient space of $W$

d) What is a basis for a vector space $V$?

A subset of vectors in $V$ that are linearly independent and vector space span $V$

e) If $V$ and $W$ are vector spaces, what is a linear transformation $T: V \to W$

A linear transformation $T: (V, \oplus_{v}, \odot_{v}) \to (W, \oplus_{w}, \odot_{w})$ is a function that assigns a unique member $w$ within $W$ to every vector $v$ within $V$, such that $T$ satisfies for all $u, v$ within $V$ and all scalar $k$ within $\mathbb{R}$: under Additivity and Homogeneity