(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Here's a statement, and I am supposed to show that it holds.

If x,y, and z are vectors such that x+y+z=0, then x and y span the same subspace as y and z.

2. Relevant equations

N/A

3. The attempt at a solution

If x+y+z=0 it means that the set {x,y,z} of vectors is linearly dependent. Because of this dependence, the vectors cannot span a subspace with dimension greater than 2.

That is, they can span subspaces with dimensions 0,1 and 2.

- If they span a subspace with dim=0, then x=y=z=0.

- If they span a subspace with dim=1, then two vectors are negative multiples of each other with the third one being the zero vector.

- If they span a subspace with dim=2, then one is a linear combination (with -1 as coefficients) of the other two.

In all these cases x and y span the same subspace as y and z.

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Any suggestions are greatly appreciated.

Thanks.

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# Homework Help: Span of subspace

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