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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

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(a) There is no need for S to be countable when discussing span. As you mentioned, span(S) is the set of all linear combinations of elements in S and in a linear combination of infinitely many elements, by definition, only finitely many coefficients are allowed to be nonzero. So essentially, it is the set of finite sums of scalar multiples of elements from S.

(b) I think you've mixed up Linear Independence and Linear Dependence. A set of vectors S is linearly independent if there is no nontrivial linear combination of elements in S which equals 0.

(c) Your idea of a subspace of a vector space is correct.

(d) Your idea of a basis is correct.

(e) I think you understand linear transformations even if you didn't fully articulate your idea.

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