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

Homework Equations

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

 

[Mark44]

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)

Homework Equations

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.

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

 

[STEMucator]

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.

 

[phys2]

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