Span and Vector Space: Understanding Vectors in Linear Algebra

In summary, the question is whether the span of three vectors in a vector space V is equal to V or if it is only a subset of V. The answer is yes to both questions, as long as the vectors are linearly independent and the vector space V is not more than 3-dimensional. If these conditions are not met, then the span of the three vectors may not be the full vector space V.
  • #1
Poetria
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Homework Statement



The question is:
if vectors v1, v2, v3 belong to a vector space V does it follow that:

span (v1, v2, v3) = V

span (v1, v2, v3) is a subset of V.[/B]

2. The attempt at a solution:

If I understand it correctly the answer to both questions is yes.
The first: the linear combinations of these three vectors fill the space V.

Am I correct? I would like to make sure if I have understood the definitions.
 
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  • #2
Poetria said:

Homework Statement



The question is:
if vectors v1, v2, v3 belong to a vector space V does it follow that:

span (v1, v2, v3) = V

span (v1, v2, v3) is a subset of V.[/B]

2. The attempt at a solution:

If I understand it correctly the answer to both questions is yes.
The first: the linear combinations of these three vectors fill the space V.

Am I correct? I would like to make sure if I have understood the definitions.

Why must the span of three vectors be the full vector space? If so, then why not the span of just two vectors?
 
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  • #3
PeroK said:
Why must the span of three vectors be the full vector space? If so, then why not the span of just two vectors?

Yes, I know, they should be linearly independent to be a spanning set. Well, we don't know if they are independent. :( I guess there may be also other vectors needed for a spanning set for the vector space V: we don't know this either. In this case the answer to the first question would be negative. But the second would hold. Have I got it?

I have read the definitions so many times that I am somewhat dizzy.
 
  • #4
Poetria said:
Yes, I know, they should be linearly independent to be a spanning set. Well, we don't know if they are independent. :( I guess there may be also other vectors needed for a spanning set for the vector space V: we don't know this either. In this case the answer to the first question would be negative. But the second would hold. Have I got it?

I have read the definitions so many times that I am somewhat dizzy.

There are two reasons why the span of ##v_1, v_2, v_3## may not be all of ##V##. If ##V## is 3-dimensional, the vectors may be linearly dependent. And, ##V## may be more than 3-dimensional in the first place.
 
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  • #5
PeroK said:
There are two reasons why the span of ##v_1, v_2, v_3## may not be all of ##V##. If ##V## is 3-dimensional, the vectors may be linearly dependent. And, ##V## may be more than 3-dimensional in the first place.

Thank you very much. Everything is clear to me now. :)
 

1. What is the difference between span and vector space?

The span of a set of vectors is the set of all possible linear combinations of those vectors. A vector space is a collection of vectors that satisfies certain properties, such as closure under addition and scalar multiplication. In other words, the span is a subset of the vector space.

2. How do you determine if a set of vectors spans a vector space?

To determine if a set of vectors spans a vector space, you can use the span test. This involves setting up a system of equations using the vectors as the variables, and solving for the constants. If the system has a solution for every vector in the vector space, then the set of vectors spans the vector space.

3. Can a single vector span a vector space?

No, a single vector cannot span a vector space. This is because a vector space must contain the zero vector, and a single vector cannot produce the zero vector through linear combinations. However, a single vector can span a one-dimensional subspace of a vector space.

4. What are the characteristics of a vector space?

A vector space must have a set of vectors, a field of scalars, and two operations - addition and scalar multiplication - that satisfy certain properties. These properties include closure, commutativity, associativity, distributivity, and the existence of an identity element and inverse elements.

5. Can a vector space contain an infinite number of vectors?

Yes, a vector space can contain an infinite number of vectors. This is because a vector space can have any number of dimensions, and each dimension can have an infinite number of vectors. For example, the vector space of all real numbers is an infinite-dimensional vector space.

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