show that this system is not a vector space

charmedbeauty

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

Show that the system S is not a vector space by showing one of the axioms is not satisfied with the usual rules for addition and multiplication by a scalar in ℝ³

S={x(ℝ³):2x₁+3x₃₂-4x₂₃=0}


2. Relevant equations



3. The attempt at a solution

The subscripts for the x's are strange I think but I guess that shouldn't make a difference.

But I'm really stuck on this I think it should be fairly easy though.

but for example if I try closure under scalar multiplication

or closure under addition I keep coming to a dead end because I don't know the value of the x's.

ie λ[2x₁+3x₃₂-4x₂₃]=λ[0]

...λ[2x₁+3x₃₂-4x₂₃]=0

I think I just need a hint in the right direction. Thanks

HallsofIvy

The way I see it, you have one major problem- what you are trying to prove is not true!

Now, please tell us the exact wording of the problem. I suspect you are misstating it.

Mark44

I don't understand what you have above. Presumably x is a vector in R³. Why are there two subscripts on some of the variables in your equation?

charmedbeauty

Ok sorry I did mistake the problem.

it really should have said 2x₁+3x³₂-4x₃²

I mistakenly read it as it was not typed out ver well and it appeared to look like weird subscript notation.

and I solved it no problems

Thanks for letting me know!