General Lin. Alg. Span Question

In summary, the answer to the question "does v1, v2, v3 span R4" is no, regardless of the values of v1, v2, and v3. This is because v1, v2, and v3 only have 3 dimensions, while R4 requires 4 dimensions. Therefore, this set of vectors cannot span R4.
  • #1
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If a question asks "does v1, v2, v3 span R4"
can i just say no, regardless of what v1 v2 v3 is
because v1 v2 and v3 is just 3 dimensions while R4 needs 4 dimensions?
 
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  • #2
hydralisks said:
If a question asks "does v1, v2, v3 span R4"
can i just say no, regardless of what v1 v2 v3 is
because v1 v2 and v3 is just 3 dimensions while R4 needs 4 dimensions?

I'm almost sure you're right on this.
 
  • #3
Thanks!
 
  • #4
Yes. If a set of vectors spans a space, then some subset of it forms a basis for the space. Since no subset of 3 vectors can span a four dimensional set, this set cannot span R4.
 

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

In general linear algebra, the span of a set of vectors is the set of all possible linear combinations of those vectors. It represents the entire space that can be created by those vectors.

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

To determine if a vector is in the span of a given set of vectors, you can use the method of Gaussian elimination or row operations to solve for the coefficients of the linear combination. If a solution exists, then the vector is in the span. If no solution exists, then the vector is not in the span.

3. Can a set of vectors span a vector space without being a basis?

Yes, a set of vectors can span a vector space without being a basis. For a set of vectors to be a basis, they must not only span the vector space, but they must also be linearly independent. A set of vectors can span a vector space without being linearly independent, in which case it would not be a basis.

4. How does the concept of span relate to linear independence?

The concept of span and linear independence are closely related. A set of vectors is linearly independent if and only if no vector in the set can be written as a linear combination of the other vectors in the set. This means that the span of a set of linearly independent vectors will be the entire vector space, whereas the span of a set of linearly dependent vectors will be a subspace of the vector space.

5. Can the span of a set of vectors change if a vector is added or removed from the set?

Yes, the span of a set of vectors can change if a vector is added or removed from the set. If the new vector is a linear combination of the original set, then the span will remain the same. However, if the new vector is not a linear combination of the original set, then the span will increase or decrease depending on the linear independence of the new vector with respect to the original set.

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