Subspaces and Span: Exploring Vector Spaces and Their Properties

In summary: It means that v is in every element of S, which is a contradiction.In summary, it seems that you are not able to solve the question.
  • #1
kbartlett
2
0
I got a small test tomorrow and i have been working throu exercises but i can't seem to solve this question:

Let V be a vector space over a field F, and let [tex]S\subset S'[/tex] be subsets of V.

a) Show that span(S) is a subspace of V.

b) Show that span(S) is a subset of Span(S').

c) Take [tex] V = R^3 [/tex] and give an example to show that it is possible that Span(S) = Span (S') even though [tex] S \subset S'[/tex] and [tex] S \neq S' [/tex].

d) Let U,W be subspaces of V. Prove that U+W is also a subspace of V.
 
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  • #2
kbartlett said:
I got a small test tomorrow and i have been working throu exercises but i can't seem to solve this question:

Let V be a vector space over a field F, and let [tex]S\subset S'[/tex] be subsets of V.

a) Show that span(S) is a subspace of V.

b) Show that span(S) is a subset of Span(S').

c) Take [tex] V = R^3 [/tex] and give an example to show that it is possible that Span(S) = Span (S') even though [tex] S \subset S'[/tex] and [tex] S \neq S' [/tex].

d) Let U,W be subspaces of V. Prove that U+W is also a subspace of V.

Go back to your definitions of subspace and span. These problems become pretty easy when you do.

A subspace is a subset of a vector space which is still closed under addition and scalar multiplication. Stated another way, any linear combination of vectors in S will be in S (or of S' in S').

The span is simply the set of linear combinations of a set of vectors. That is, you can take any finite number of vectors from S, make them as long or as short as you want, and add them together and the result is always in Span(S).

For the proof of (d), I'm not sure of your use of + in U+W. U and W are sets, which we don't usually have an addition operation for. Maybe you meant U union W or maybe you're using a notation I'm just not familiar with or something.

(Also, use the [tex ] tag, not [math ] on these boards. The latter doesn't work.
 
  • #3
Welcome to PF kbartlett. Instead of "math", you may want to try "tex" :wink:

It seems to me that a) and b) are the same question, if you replace S by V and S' by S in b).
Try writing down an element from span(S) which is not in V, and use the axioms of a vector space to derive a contradiction.

For c: try to make S a set that spans the plane [tex]\mathbb R^2[/tex]. Do you see how to construct S'?

For d: check the axioms of a vector space.

If you need more specific advise, please post some workings.
 
  • #4
Tac-Tics said:
For the proof of (d), I'm not sure of your use of + in U+W. U and W are sets, which we don't usually have an addition operation for. Maybe you meant U union W or maybe you're using a notation I'm just not familiar with or something.

As U and W are subspaces of a vector space (which has an addition [itex]+_V[/itex]), I think it is not unreasonable to assume that
[tex]U + W = \{ u +_V w \mid u \in U, w \in W \}[/tex]
 
  • #5
Thanks i think I've got the answer to part d), but I am still stuck on the rest.
 
  • #6
Suppose that I tell you that some vector, v, is in Span(S). What does this mean?
 

1. What is the difference between span and subspace?

Span and subspace are both mathematical concepts used in linear algebra. The main difference between them is that span refers to the set of all possible linear combinations of a given set of vectors, while subspace refers to a vector space that satisfies certain properties. In other words, span is a set of vectors, while subspace is a space of vectors.

2. How do you determine if a vector is in the span of a given set of vectors?

A vector is in the span of a given set of vectors if it can be expressed as a linear combination of those vectors. This means that the vector can be written as a sum of scalar multiples of the vectors in the set. To determine this, you can use the method of Gaussian elimination or row reduction to see if the vector can be written as a linear combination of the given vectors.

3. Can a subspace contain non-linear combinations of vectors?

No, a subspace can only contain linear combinations of vectors. This means that the vectors in a subspace must be closed under scalar multiplication and vector addition. If a subspace contains non-linear combinations of vectors, it would violate these properties and therefore would not be considered a subspace.

4. How does the dimension of a span relate to the dimension of a subspace?

The dimension of a span is equal to the number of linearly independent vectors in the set, while the dimension of a subspace is equal to the number of basis vectors in the subspace. Since the basis vectors of a subspace are also linearly independent, the dimension of a subspace is also equal to the dimension of its span.

5. Can a set of vectors span a subspace of a higher dimension?

Yes, a set of vectors can span a subspace of a higher dimension. This is because a subspace can be spanned by more than one set of basis vectors, and these basis vectors can have a higher dimension than the original set of vectors. For example, a set of two vectors in 3-dimensional space can span a subspace of 2 dimensions.

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