# Do the vectors u = (5,1,3) and v = (2,3,6) belong to span(S)?

• phys2
In summary, to determine if vectors u and v belong to the span of set S, they must be able to be expressed as linear combinations of the vectors in S. This can be checked by setting up and solving a system of equations using matrix form. If the system is dependent, the vector is not in the span of S. If the system is independent, the vector is in the span of S.
phys2

## Homework Statement

The problem is : Let S = [ (1,-1,3) , (-1,3, -7) , (2,1,0) ]. Do the vectors u = (5,1,3) and v = (2,3,6) belong to span(S)

## The Attempt at a Solution

So span means that I could take linear combinations of u and v and they should end up giving (1,-1,3) , (-1,3,-7) and (2,1,0). Right?

I could take x [5 1 3 ] + y [ 2 3 6 ] = [1 -1 3] or [-1 3 -7 ] or [2 1 0 ] (btw i meant to write [ 5 1 3] as a column matrix but I am not sure of how to using Latex. So anyway, is what I am trying to do correct? Is that what it means for the vectors to span S?

Thanks

phys2 said:

## Homework Statement

The problem is : Let S = [ (1,-1,3) , (-1,3, -7) , (2,1,0) ]. Do the vectors u = (5,1,3) and v = (2,3,6) belong to span(S)

## The Attempt at a Solution

So span means that I could take linear combinations of u and v and they should end up giving (1,-1,3) , (-1,3,-7) and (2,1,0). Right?
No, it's the other way around.

Span(S) is the set of all linear combinations of the vectors in S. u is in Span(S) if there are constants a, b, and c for which a(1, -1, 3) + b(-1, 3, -7) + c(2, 1, 0) = u.

Similarly for v.
phys2 said:
I could take x [5 1 3 ] + y [ 2 3 6 ] = [1 -1 3] or [-1 3 -7 ] or [2 1 0 ] (btw i meant to write [ 5 1 3] as a column matrix but I am not sure of how to using Latex. So anyway, is what I am trying to do correct? Is that what it means for the vectors to span S?

Thanks

Put your vectors from S into matrix form, augmenting them with either u or v ( You'll have to do both at some point so pick one at a time ).

Solve the corresponding system and check if the following system is linearly independent or dependent.

If the system is dependent for your choice of u or v, then you can conclude that the vector is not in the span of your set. Otherwise if your system is independent, you can exhibit a unique solution for your system implying that your vector IS in the span of your set.

No, it's the other way around.

Span(S) is the set of all linear combinations of the vectors in S. u is in Span(S) if there are constants a, b, and c for which a(1, -1, 3) + b(-1, 3, -7) + c(2, 1, 0) = u.

Similarly for v.

Ahh I see thanks

Solve the corresponding system and check if the following system is linearly independent or dependent.

Yes, it works out...thanks

## 1. What is the concept of spanning in linear algebra?

In linear algebra, "spanning" refers to the ability of a set of vectors to reach or cover the entire vector space by linear combinations of those vectors.

## 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 "spanning test" which involves checking if every vector in the vector space can be written as a linear combination of the given set of vectors.

## 3. What is the importance of spanning in linear algebra?

The concept of spanning is important in linear algebra because it helps us determine the basis of a vector space and understand the relationships between vectors in a given set.

## 4. Can a set of vectors span more than one vector space?

No, a set of vectors can only span one vector space. This is because the definition of spanning requires that the set of vectors covers or reaches the entire vector space, not just a part of it.

## 5. Is it possible for a set of vectors to span a vector space if one of the vectors can be written as a linear combination of the others?

Yes, it is still possible for a set of vectors to span a vector space even if one of the vectors can be written as a linear combination of the others. This is because the spanning test only requires that every vector in the vector space can be written as a linear combination of the given set of vectors, not necessarily that each vector has a unique representation.

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