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
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
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.
If we take the span of the three vectors , , and over we get the vector space . Can you see why?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
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.
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?
Because it is an independent span, therefore a basis for R3, right?
Ohhhh, thank you, I understand it now. Finally can make the connection to the fact that V can be infinite but spanV is finite.