- #1

Dafe

- 145

- 0

## Homework Statement

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.

## Homework Equations

N/A

## 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.