Some technical questions about spans and bases

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In summary, the conversation discusses technical questions about spans and bases, and the answers to the questions are as follows: 1) Yes, 2) Yes, 3) Yes, 4) No, 5) Yes, and 6) Yes. The definition of "span" implies that there is only one space spanned by a given set S, and the span of S contains every subspace of V as a subset. However, for a strict mathematical formulation, it is important to specify that everything is a part of a vector space W, which can be identical to V.
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Bipolarity
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I have some technical questions about spans and bases which my textbook really does not cover very well. I would appreciate answers to them. They are not textbook problems, merely specific quesitons relating to the definitions of "span" and "basis".

1) If span(S) = V, need the elements of S be in V?
2) If S is a basis for V, need the elements of S be in V?
3) If S spans V, does S span every subspace of V?
4) If S is a basis for V, is S a basis for every subspace of V?
5) If S is linearly independent, is every subset of S also linearly independent?
6) If S is linearly dependent, is every superset of S also linearly dependent?

If I am not wrong, the answers to the questions are:
1) Yes
2) Yes
3) Yes
4) No
5) Yes
6) Yes

But I would like explicit confirmation.
All help is appreciated. Thanks.

BiP
 
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  • #2
All correct. For a strict mathematical formulation, you should let everything be a part of a vector space W (can be identical to V, but then questions 1 and 2 are meaningless).

3) If S spans V, does S span every subspace of V?
Yes
The span of S contains every subspace of V as subset.
 
  • #3
I disagree about (3).

If S spans V and W is a subspace of V, then the elements of S may not all be in W, so "S spans W" might contradicts the (correct IMO) answer to (1).

Also the definition of "span" in http://en.wikipedia.org/wiki/Linear_span implies that there is only ONE space that is spanned by a given set S - namely, the intersection of all subspaces of V that contain (all the vectors in) S.
 

FAQ: Some technical questions about spans and bases

1. What is the definition of a span in linear algebra?

A span is the set of all possible linear combinations of a given set of vectors. It represents the space that can be created by scaling and adding those vectors together.

2. How is the dimension of a span determined?

The dimension of a span is equal to the number of linearly independent vectors in the set. It can also be described as the minimum number of vectors needed to create the span.

3. What is the difference between a basis and a span?

A span is the set of all possible linear combinations of a given set of vectors, while a basis is a set of linearly independent vectors that can create the span. In other words, a basis is a subset of the span that is both linearly independent and spans the same space.

4. Can a span have more than one basis?

Yes, a span can have multiple bases. This is because there can be more than one set of linearly independent vectors that can create the same span. However, all bases for a given span will have the same number of vectors, which is equal to the dimension of the span.

5. How do you find a basis for a given span?

To find a basis for a given span, you need to find a set of linearly independent vectors that can create the span. This can be done through methods such as Gaussian elimination or finding the null space of the matrix representing the span.

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