how to determine if a set of vectors span a space

mathmathmad

1. The problem statement, all variables and given/known data

How to determine if a set of vectors span a space in general?
say, V=R^n and you're given a few vectors and asked to determine if they span the space..
how do you do that?

2. Relevant equations



3. The attempt at a solution

Mark44

A set S of vectors spans V iff every vector in V can be written as a linear combination of vectors in S.

Just to make this a little less abstract, suppose V = R³, and that S = {<1, 0, 1>, <0, 2, 5>}. Does this set of vectors span V?

mathmathmad

I'm not quite sure with my way of finding the answer, which is NO i.e. do not span

would it be okay if you show me your working based on this example? :)

Mark44

You tell me why you think this set doesn't span R³.

mathmathmad

erm, I wrote it as augmented matrix

1 0 | 1 0 0
0 2 | 0 1 0
1 5 | 0 0 1

and start to reduce it to "reduced row echelon form" (that's why I started another thread before this asking about RRE form because I'm unsure how this works)

and on the 3rd row, i get 0 0 | 0 1/2 1/5 (inconsistent, so do not span?)

:( I'm looking for another way of determining the spanning set
this is what I get from google-ing O_O

Mark44

There's a much simpler way to answer the question, that can be done with no computation. Answer the following questions and it will be obvious to you.

What's the dimension of R³ (i.e., dim(R³))?
How many vectors does it take to span R³?
How many vectors are there in S?
Does S span R³?

Going back to the work you did, you have a lot of extra stuff that doesn't make any sense to me. A given set of vectors spans R³ if any vector in R³ is some linear combination of the vectors in the set. IOW, for any vector <x, y, z>, there is a solution for the constants a and b in this equation:
a<1, 0, 1> + b<0, 2, 5> = <x, y, z>

Setting this up as an augmented matrix gives you this:

1 0 | x
0 2 | y
1 5 | z

After row reduction, I get j
1 0 | x
0 1 | y/2
0 0 | z-x -5y/2

The first two rows say that a = x and b = y/2, but the bottom row says that 0a + 0b = z - x - 5y/2. This last equation is saying that the system of equations has a solution only if z - x -5y/2 = 0. IOW, for some vectors <x, y, z> there is no solution.