The Question concerns "Linear Algebra [ Singh & Nair ] 2018" where this Exercise 1.9.1 (Page 47) is given :
Is there any logical mistake in this proof? I tried $$x+x+y+y=2x+2y=2(x+y)=(1+1)(x+y)=x+y+x+y$$
$$\therefore x+x+y+y=x+y+x+y$$ $$ \implies -x+x+x+y+y-y=-x+x+y+x+y-y \implies x+y=y+x $$
I can't see what is wrong with this... Is this wrong because for field of characteristic $2$, it will become $0$ and hence can't proceed with a proof or is there anything else? kindly help
There are various ways to make the Definition for Vector Spaces , in various text books by various Authors & ETC.
The Question concerns "Linear Algebra [ Singh & Nair ] 2018" & hence , to make sense of the given Exercise 1.9.1 (Page 47) , we have to use the Axioms given there.
The Issue lies with Axiom 3 (Page 2)
"There exists an element in $V$ , called a zero vector , denoted by $0$ , such that $\forall x \in V , x+0=x$"
& with Axiom 4 (Page 3)
"For each $x \in V$ , there exists an element in $V$ , denoted by $−x$ , and called an additive inverse of $x$ , such that $x+(−x)=0$"
Compare that with "Terence Tao" with this article :
https://www.math.ucla.edu/~tao/resource/general/121.1.00s/vector_axioms.html
"there are a number of ways to phrase these axioms, and different books will do this differently, but they are all equivalent"
"$0+x = x+0 = x$"
"$(-x) + x = x + (-x) = 0$"
So , what is the Issue ?
(A) While Tao assumes (axiomatically) that $0$ commutates with very other element , Singh+Nair do not.
(B) While Tao assumes (axiomatically) that every element commutates with inverse element , Singh+Nair do not.
Proof uses $0+P=P$ , which is not yet known , we know $P+0=P$.
Proof uses $[-x]+x=0$ , which is not yet known , we know $x+[-x]=0$.
When we overcome that Issue , Proof will be right.
Currently , it is incomplete.
PUTTING IT IN OTHER WORDS :
(A) We can not cancel $x$ from left , since it will use $[-x]+x=0$ which is unknown at the moment.
(B) Even with that , we will get "$[-x]+x+[[other-terms]]=0+[[other-terms]]$" & we can not make it "$0+[[other-terms]]=[[other-terms]]$" , since it will use $0+x=x$ which is unknown at the moment.
POST SCRIPT :
I see that user "blargoner" has made such Observations earlier.
I took the approach to compare with well-known Authors & check what might be missing , coming to the Same Conclusion.