# spanning

• Nov 13th 2009, 08:17 AM
alexandrabel90
spanning
does (0,0,0,0) span {( 2,1,0,3), (3,-1,5,2), ( -1, 0 2,1)}

i tried to use Gaussian elimination and i found that the the coefficients for these 3 vectors to be expressed as a combination of the zero vector have to be all 0.

this should mean that there exist no linear combi of these 3 vectors that will give the zero vector and hence the zero vector does not span {( 2,1,0,3), (3,-1,5,2), ( -1, 0 2,1)} right?

but my answer states that it spans {( 2,1,0,3), (3,-1,5,2), ( -1, 0 2,1)}....

did i do sth wrongly?
• Nov 13th 2009, 08:48 AM
HallsofIvy
Quote:

Originally Posted by alexandrabel90
does (0,0,0,0) span {( 2,1,0,3), (3,-1,5,2), ( -1, 0 2,1)}

I don't understand the question.A set of vectors "spans" the subspace of all linear combinations of vectors in the set. {(2,1,03), (3,-1,2), (-1,0,2,1)} is a set of vectors, not a subspace so it makes no sense to ask if any set of vectors "spans" it. You could ask if a given set of vectors spans the same subspace that {(2,1,03), (3,-1,2), (-1,0,2,1)}, but then the answer is trivally no. Any "linear combination" of the single vector (0,0,0,0) is just (0,0,0,0) itself.

Quote:

i tried to use Gaussian elimination and i found that the the coefficients for these 3 vectors to be expressed as a combination of the zero vector have to be all 0.
Again that makes no sense. Do you mean to say that for the zero vector to be expressed as a combination of these vectors (the reverse of what you said above), the coefficients must be 0? That would be saying that these vectors are independent.

Quote:

this should mean that there exist no linear combi of these 3 vectors that will give the zero vector and hence the zero vector does not span {( 2,1,0,3), (3,-1,5,2), ( -1, 0 2,1)} right?
Once again, the zero vector does not span anything except itself. If you are asking if the span of this set of vectors includes the zero vector, the answer it trivially "yes". Just take the coefficients all equal to 0. That's a linear combination.

Quote:

but my answer states that it spans {( 2,1,0,3), (3,-1,5,2), ( -1, 0 2,1)}....

did i do sth wrongly?
Please write the problem exactly as it is given.
• Nov 13th 2009, 10:02 AM
alexandrabel90
sorry!

the question was

let v1 = (2,1,0,3), v2=(3,-1,,2), v3 = (-1,0,2,1). is (0,0,0,0) in the span {v1,v2,v3}?
• Nov 13th 2009, 10:04 AM
alexandrabel90
from your reply above, i suppose the answer to trivially yes..but if all the coefficients are 0, is that still as it is a linear combination of that set of vectors?
• Nov 13th 2009, 10:11 AM
alexandrabel90
when i say that a vector (A) is in the span of a set of vectors( B) , does it mean that the vector A can be expressed as a linear combination of the set of vectors( B)?

thanks!
• Nov 14th 2009, 02:08 AM
HallsofIvy
The question, as you posted it is pretty much the exact opposite of the question you originally posted! You are asking if the 0 vector is in the span of this set of vectors and originally you had asked if the 0 vector spanned the set.

In any case, yes, the problem is asking if the 0 vector can be written as a linear combination of the vectors in the set. And a linear combination is a sum of any numbers times those vectors- even 0 and even all 0s. The 0 vector is trivially in the span of any set of vectors.