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## Homework Statement

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?

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- Thread starter mathmathmad
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- #1

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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

Mark44

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## Homework Statement

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?

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 = R

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

- #4

Mark44

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You tell me why you think this set doesn't span R^{3}.

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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

- #6

Mark44

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What's the dimension of R

How many vectors does it take to span R

How many vectors are there in S?

Does S span R

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 R

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.

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