Is V in Span(S)

• Mar 22nd 2013, 10:25 PM
Cotty
Is V in Span(S)
Hey,

I have a question that I'm not sure how to approach at all. The question is:

Let

S={[[-1],[2],[1]] , [[3],[1],[2]] , [[1],[5],[4]] , [[-6],[5],[1]]}

and

v=[[-5],[3],[0]]

Is v in span(S)?

So i know that a set of vectors ie. v, spans a subspace, S, if every vector in S can be written as a linear combination of vectors in v.

• Mar 22nd 2013, 11:12 PM
Re: Is V in Span(S)
Alright, let's us definition as you laid out. If v is a linear combination of vectors in S, then there exists constants $c_i$ such that

$c_1(-1,2,1) + c_2(3,1,2) + c_3(1,5,4) + c_4(-6,5,1) = (-5,3,0)$

by comparison we get the simultaneous system of equations

$-c_1 + 3c_2 + 1c_3 + -6c_4 = -5$
$2c_1 + c_2 + 5c_3 + 5c_4 = 3$
$1c_1 + 2c_2 + 4c_3 + 1c_4 = 0$

You can solve this system using a matrix. If such $c_i$ exist, then you have show v is in a span of S. If no such $c_i$ exist then v is not in the span of S.
• Mar 22nd 2013, 11:31 PM
Cotty
Re: Is V in Span(S)
Quote:

Alright, let's us definition as you laid out. If v is a linear combination of vectors in S, then there exists constants $c_i$ such that

$c_1(-1,2,1) + c_2(3,1,2) + c_3(1,5,4) + c_4(-6,5,1) = (-5,3,0)$

by comparison we get the simultaneous system of equations

$-c_1 + 3c_2 + 1c_3 + -6c_4 = -5$
$2c_1 + c_2 + 5c_3 + 5c_4 = 3$
$1c_1 + 2c_2 + 4c_3 + 1c_4 = 0$

You can solve this system using a matrix. If such $c_i$ exist, then you have show v is in a span of S. If no such $c_i$ exist then v is not in the span of S.

So when I row reduced that matrix, I ended up with [[1,0,2,3,2],[0,1,1,-1,-1],[0,0,0,0,0]]

Seeing as this means that there are infinitely many solutions does this mean that v is NOT in span(S)?
• Mar 22nd 2013, 11:35 PM
Re: Is V in Span(S)
No, the solution for $c_i$ need not be unique in order for v to be in the span (choose one linear combination and you're done). In fact, you are bound to have at least one free variable since you have four variables and three equations.
• Mar 22nd 2013, 11:47 PM
Cotty
Re: Is V in Span(S)
Ok, so because there is a solution, even if it is infinitely many, we can say that v is in span s? Thank you very much for your time, you have been a great help by the way! So how could v be expressed as a linear combination of the vectors from s then? Am i correct in saying that it would be:

if we let c_3=t and c_4=s

(2-2t-3s)(-1,2,1)+(-1-t-s)(3,1,2)+t(1,5,4)+s(-6,5,1)=(-5,3,0)
• Mar 22nd 2013, 11:50 PM
Re: Is V in Span(S)
This looks correct. Well done.
• Mar 22nd 2013, 11:55 PM
Cotty
Re: Is V in Span(S)
Thank you so much for your time!! I was dreading trying to figure out how to do this question, you have been such a great help
• Mar 22nd 2013, 11:57 PM