Proof of a theorem about spanning sets

[jwqwerty]

Homework Statement

Let W be a subspace of R^n and let A be a subset of R^n. Then A spans W if and only if <A> = W (<A> is the set of all vectors in R^n that are dependent on A). Prove it.

Homework Equations

The Attempt at a Solution

Ok the book goes likes this;
First, it proves that if A spans W, then <A>=W
It says that every vector in W is dependent on A, so W⊆<A>. But i do not get this part. Why is this true?

 

[sunjin09]

<A> is the set of all vectors in R^n that are dependent on A
It says that every vector in W is dependent on A, so W⊆<A>. But i do not get this part. Why is this true?

Do you see it now?

 

[HallsofIvy]

What does "dependent on A" mean?