1. ## Subsets generating supspaces

Show that two subsets A and B of a vector space V generate the same subspace if and only if each vector in A is a linear combination of vectors in B and vice versa.

(a) if two subsets A and B of a vector space V generate the same subspace, then each vector in A is a lin. comb. of vectors in B.
I am thinking by contradiction: Suppose subsets A and B of a vector space V don't generate the same subspace and a in A is a lin. comb. of B.

$\displaystyle \mathbf{a}_i=\sum_{i\in I}\lambda_i\mathbf{b}_i, \ \lambda_i\in\mathbb{F}$
From this, it seems trivial that A generates the same subspace. However, I could be wrong or I just don't know how to articulate it from here

Don't worry about answering the other direction. I am focusing on this one now.

2. ## Re: Subsets generating supspaces

Originally Posted by dwsmith
Show that two subsets A and B of a vector space V generate the same subspace if and only if each vector in A is a linear combination of vectors in B and vice versa.

(a) if two subsets A and B of a vector space V generate the same subspace, then each vector in A is a lin. comb. of vectors in B.
I am thinking by contradiction: Suppose subsets A and B of a vector space V don't generate the same subspace and a in A is a lin. comb. of B.

$\displaystyle \mathbf{a}_i=\sum_{i\in I}\lambda_i\mathbf{b}_i, \ \lambda_i\in\mathbb{F}$
From this, it seems trivial that A generates the same subspace. However, I could be wrong or I just don't know how to articulate it from here

Don't worry about answering the other direction. I focusing on this one now.
I think you're making it more difficult than it is. Suppose that $\displaystyle \text{span }A=\text{span }B$, then in particular, since $\displaystyle A\subseteq\text{span }A$ we of course have that $\displaystyle A\subseteq\text{span }B$ which by definition means each element of $\displaystyle A$ is a linear combination of elements of $\displaystyle B$, etc.

3. ## Re: Subsets generating supspaces

Originally Posted by Drexel28
I think you're making it more difficult than it is. Suppose that $\displaystyle \text{span }A=\text{span }B$, then in particular, since $\displaystyle A\subseteq\text{span }A$ we of course have that $\displaystyle A\subseteq\text{span }B$ which by definition means each element of $\displaystyle A$ is a linear combination of elements of $\displaystyle B$, etc.
How can we go from subsets A and B to speaking about their spans?

4. ## Re: Subsets generating supspaces

Originally Posted by dwsmith
How can we go from subsets A and B to speaking about their spans?
Saying that they generate the same subspace is equivalent to saying they have equal spans.

5. ## Re: Subsets generating supspaces

Then for the other direction assuming the contrary would be the way to go? Since we are assuming the contrary, there exists an a in A s.t. $\displaystyle a\neq\sum\lambda_ib_i$. Therefore, the span A > span B which is a contradiction.

Correct?