Proving Equivalence of Subspaces: x+y+z=0

[Dafe]

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.

 

[HallsofIvy]

I think you are missing the point of the question.

Let v be in the subspace spanned by x and y. Then v= ax+ by for some numbers a and b. But x+ y+ z= 0 so x= -y- z. v= a(-y- z)+ by= (b-a)y+ (-a)z. That is, v is a linear combination of y and z and so is in the span of y and z.

That proves that the subspace spanned by x and y is a subspace of the span of y and z. I will leave it to you to show that the span of y and z is a subspace of the span of x and y.

Let v be in the subspace spanned by y and z, Then v= ...

 

[Dafe]

Ah, I certainly missed the point of the question!

Let v be in the subspace spanned by y and z.
Then v=ay+bz for some numbers a and b.
But x+y+z=0 so z=-x-y.
v=(a-b)y+(-bx), that is, v is a linear combination of y and x and so is in the span of y and x.
This proves that the subpsace spanned by y and z is a subspace of the span of y and x.

We have now proved that the subspace spanned by x and y is the same as the one spanned by y and z.

Thank you!