Verifying $(x_1,y_1) + (x_2,y_2) = (2x_1 - 2y_1,-x_1+y_1)$ is not vector space

Question

I need to prove that this is not a subspace:

$$(x_1,y_1) + (x_2,y_2) = (2x_1 - 2y_1,-x_1+y_1)$$

(and with the multiplication definied in a way that I will not write because the sum already fails)

$$(x_2,y_2) + (x_1,y_1) = (2x_2 - 2y_2,-x_2+y_2)$$ I noted that in the first part of the vectors, if we have:

$$x_1\neq x_2, y_1\neq y_2$$ then the two vectors are different. I picked, then: $$x_1 = 2, x_2 = 1, y_1 = 3, y_2 = 4$$ and made the sum:

$$(2,3)+(1,4) = (2.2-2.3, -2+3) = (-2,1)\\(1,4)+(2,3) = (2.1-2.4,-1+4) = (-6,3)$$

Since I ended with two different vectors, the sum is not commutative, therefore this is not a vector space.

Is my reasoning right?

Answer 1

It is true, yet there are easier proofs. For example, let $(\phi_1, \phi_2)$ be a zero vector. Then

$$(1,0) = (1,0) + (\phi_1, \phi_2) = (2, -1)$$

which is a contradiction.