# proving associative property of any two vectors

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• Jan 21st 2013, 10:00 AM
bonfire09
proving associative property of any two vectors
Problem:Prove that any two ways of associating a sum of any number of vectors give the same sum. (Hint. Use induction on the number of vectors.)

Proof: Base Case: Let there be three vectors namely v1,v2,v3. We see that (v1+v2)+v3=v1+(v2+v3).

Inductive Case: Let k be the number of vectors where k∈N. We want to assume that for v1...vkthat we could associate them any two ways and get the same sum. We want to show for k+1 vectors that the same thing holds.

Case 1: Odd number of k+1 vectors
( (v1+v2)+...+(vk-1+vk))+vk+1 =(v1+ (v2+v3)+...+(vk+Vk+1))=v1+ ((v2+v3)+...+(vk+Vk+1))

Case 2: Even Number of k+1 vectors
((v1+v2)+...+(vk-2+vk-1)+vk)+vk+1=(v1+(v2+v3)+..+(vk-1+vk))+vk+1=(v1+v2)+...+(vk+vk+1)=v1+((v2+v3)+..+(vk+vk+1))

Hence by induction it holds for k+1 vectors.

Did I do it correctly?
• Jan 21st 2013, 06:10 PM
Deveno
Re: proving associative property of any two vectors
you've only shown 3 out of the many, many cases for k odd, and 4 cases for k even. this particular proof is a pain, because the number of cases grows rapidly as k does.

let's say S is a sum (associated arbitrarily) of k+1 vectors. we want to show that S = ((...(v1+v2)+v3)+...+vk)+vk+1. we will assume this is true for any sum of j vectors with: 3 ≤ j ≤ k.

we can write:

S = A+B, for two sums A and B, and the largest either A or B can be is a sum of k vectors.

applying our induction hypothesis to B we have:

B = ((...(vj+vj+1)+vj+2)+...+vk)+vk+1, for some vj.

applying our induction hypothesis to A, we have:

A = ((...(v1+v2)+v3)+...+vj-2)+vj-1.

if we let C = ((...(vj+vj+1)+...+vk-2)+vk,

we have A+B = A+(C+vk+1) = (A+C)+vk+1.

applying the induction hypothesis to A+C, we have:

A+C = ((...(v1+v2)+v3+...+vk-1)+vk,

hence S = (A+C)+vk+1 = ((...(v1+v2)+v3)+...+vk)+vk+1, as desired.

(the problem with your proof is that association can occur with more than "adjacent pairs", you can have sums like:

((v1+v2)+v3)+(v4+v5) or

(v1+(v2+v3)+v4)+v5 and so on).
• Jan 21st 2013, 06:51 PM
bonfire09
Re: proving associative property of any two vectors
oh ok thanks. this makes alot of sense now. i didn't even think of that.