1. ## span and basis

hello!

could someone explain to me the difference between span and basis?

they seem to be the same thing to me:/

2. Originally Posted by alexandrabel90
hello!

could someone explain to me the difference between span and basis?

they seem to be the same thing to me:/
A basis is a spanning set of smallest dimension.

3. Originally Posted by alexandrabel90
hello!

could someone explain to me the difference between span and basis?

they seem to be the same thing to me:/
Uh, no! Span and basis are are completely different concepts, but they are related, of course: The span of a set of vectors (which, let it be noted, need not be linearly-independent) is a vector space (typically a subspace of the vector space under consideration), namely the set of all linear combinations of (finitely many) vectors from that set; whereas a basis of a vector space is a set of linearly-independent vectors, the span of which is the entire vector space under consideration.

4. uh, yes, Mr. fantastic is correct (not to mention fantastic!). A basis for a space is a set that both spans the space and is independent. Given any spanning set, if it not already independent, there will exist a subgroup that is and thus is a basis. That is why a basis is "a smallest spanning set". (I'm not sure I would have used the term 'dimension'.)

You appear ("typically a subspace of the vector space under consideration" for span and "the entire vector space under consideration") to be asserting that a subspace cannot have a "basis"- which is not true.

5. Originally Posted by HallsofIvy
uh, yes, Mr. fantastic is correct (not to mention fantastic!). A basis for a space is a set that both spans the space and is independent. Given any spanning set, if it not already independent, there will exist a subgroup that is and thus is a basis. That is why a basis is "a smallest spanning set". (I'm not sure I would have used the term 'dimension'.)

You appear ("typically a subspace of the vector space under consideration" for span and "the entire vector space under consideration") to be asserting that a subspace cannot have a "basis"- which is not true.
I am only asserting that a span is not a basis: a span is a subspace, which is never a set of linearly-independent vectors (which is required of a basis).
Of course, any vector space, in particular every subspace of a vector space (thus even span S), has a basis.
Please clearly distinguish my saying that "span S is not a basis" (which is true, I claim) from my saying "span S does not have a basis" (which is not true, and which is not what I have claimed).
Being and having are rather different concepts: you have a computer (I presume), but it does not follow from this that you are a computer.

6. Originally Posted by HallsofIvy
uh, yes, Mr. fantastic is correct (not to mention fantastic!). A basis for a space is a set that both spans the space and is independent. Given any spanning set, if it not already independent, there will exist a subgroup that is and thus is a basis. That is why a basis is "a smallest spanning set". (I'm not sure I would have used the term 'dimension'.)

You appear ("typically a subspace of the vector space under consideration" for span and "the entire vector space under consideration") to be asserting that a subspace cannot have a "basis"- which is not true.
Yes, using dimension was a bit rash of me as the dimension of each is the actually same, but not the size.