# Is my proof of Commutativity of addition in a Vector space correct?

Adding vectors is commutative because adding coordinates is commutative and vector addition is merely two applications of that same law vectors $$A$$, $$B$$

$$A + B = (\overrightarrow{a_1,b_1}) + (\overrightarrow{a_2, b_2}) = (a_1 + b_1, a_2 + b_2) + (c_1 + d_1, c_2 + d_2)$$

which implies Commutativity be default due to the fact that we are merely adding real numbers.

Edit: Mistook commutativity for associativity!

• I am seriously confused. What are $A,B$? What exactly are you trying to show? – BigbearZzz Dec 23 '18 at 4:51
• associativity of vector addition operation -- axiom of vector space – Just do it Dec 23 '18 at 4:52
• Just for context this problem is from "A Survey of Modern Algebra" - Garrett Birkoff, and Saunders Mac Lane, p.169. – PossiblyDakota Dec 23 '18 at 5:10
• Not sure if it's not showing everything on my phone, but it seems like you have some c's and d's coning out of nowhere; you have to show $(A+B)+C=A+(B+C)$, so you should pick a side, start from it, and arrive at the other. – Ovi Dec 23 '18 at 5:16
• @PossiblyDakota Commutativity of addition is also taken as an axiom... Can you quote the problem from "A Survey of Modern Algebra"? – Dair Dec 23 '18 at 5:23

Your proof is confusing because $$c_1, c_2, d_1$$, and $$d_2$$ appear out of nowhere. But I think your general idea is correct. The proof is as follows.
The commutative property of two-dimensional real vectors is: For all two-dimensional real vectors $$a = (a_1, a_2), b = (b_1, b_2)$$, we must have $$a + b = b + a$$. This is true since
$$a + b = (a_1 + b_1, a_2 + b_2) = (b_1 + a_1, b_2 + a_2) = b + a$$