Thread: Subspace, span and spanning set

1. Subspace, span and spanning set

I have an exam tomorrow and I can't tell the difference between the 3 definitions .

Say V is a space over F.

Subspace S is:
1) a subset of V
2) a space itself over F

Span(S) is:
1) a linear combination of elements in S
2) a space itself over F

Someone told me the other day that if S is a subspace, then S=span(s). Why? I can't tell the difference between span(S) and S.
If span(s) a subspace, what is it's superset - S or V? Neither?

And what is a spanning set and how it connects to span(S)?

I'd much appreciate a concrete example, thanks

2. Originally Posted by Rita.g
I have an exam tomorrow and I can't tell the difference between the 3 definitions .

Say V is a space over F.

Subspace S is:
1) a subset of V
2) a space itself over F

Span(S) is:
1) a linear combination of elements in S
2) a space itself over F

Someone told me the other day that if S is a subspace, then S=span(s). Why? I can't tell the difference between span(S) and S.
If span(s) a subspace, what is it's superset - S or V? Neither?

And what is a spanning set and how it connects to span(S)?

I'd much appreciate a concrete example, thanks
In your second definition - the definition of span - S is not necessarily a subspace. It is just some subset of V.

Span(S) is always a subspace, and if S is a subspace then span(S)=S holds.

3. Oh, I see.
So if I'm given a subset S of space V and I'm asked to find if it's a subspace - Is it enough to find Span(S) and show if it's equal or not to S to define S as a subspace?

What is actually the purpose of span? What is it used for? I understand the term space and subspace, and I can't seem to connect span to any of those. The definition of span is like Latin to me heh
I understand the words but not the sentence

4. Originally Posted by Rita.g
Oh, I see.
So if I'm given a subset S of space V and I'm asked to find if it's a subspace - Is it enough to find Span(S) and show if it's equal or not to S to define S as a subspace?
In essence, yes. When you show that something is a subspace you show that it is closed under some conditions and conclude that it is a subspace - you are just showing that if we take the span it is contained in the subset.

What is actually the purpose of span? What is it used for? I understand the term space and subspace, and I can't seem to connect span to any of those. The definition of span is like Latin to me heh
I understand the words but not the sentence
If we take the span of the three vectors $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ over $\mathbb{R}$ we get the vector space $\mathbb{R} \times \mathbb{R} \times \mathbb{R}$. Can you see why?

Span's are useful because we can write down the vector spaces in a concise manner - we do not need to write down every single element and how they interact with one another if we just write down those elements which span the space.

5. Originally Posted by Swlabr
In essence, yes. When you show that something is a subspace you show that it is closed under some conditions and conclude that it is a subspace - you are just showing that if we take the span it is contained in the subset.
Just making sure - Not necessarily equal, but contained? For any subset S of V - S is a subspace if Span(s) is contained in S?

Originally Posted by Swlabr
If we take the span of the three vectors $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ over $\mathbb{R}$ we get the vector space $\mathbb{R} \times \mathbb{R} \times \mathbb{R}$. Can you see why?
Because it is an independent span, therefore a basis for R3, right?

Originally Posted by Swlabr
Span's are useful because we can write down the vector spaces in a concise manner - we do not need to write down every single element and how they interact with one another if we just write down those elements which span the space.
Ohhhh, thank you, I understand it now. Finally can make the connection to the fact that V can be infinite but spanV is finite.

6. Originally Posted by Rita.g
Just making sure - Not necessarily equal, but contained? For any subset S of V - S is a subspace if Span(s) is contained in S?
Yes, but it is perhaps an overly formal way of looking at it.