Is V a vector space over the field of real numbers with the following operations?

Question

$V=R^2$, with operations:

addition ($\oplus$): $(x_1, y_1) \oplus (x_2, y_2) = (x_1 + x_2, y_1 + y_2)$

multiplication($\circ$): $c \circ (x, y) = (cx, y)$

This is exercise II.1.4 from Hoffman-Kunze's Linear Algebra book. I was able to show that all the axioms are satisfied. Can someone confirm if that is correct?

$\textbf{EDIT}$: I made a mistake in checking one of the distributive properties. It is not a vector space.

Answer 1

Distributivity of multiplication over addition requires that

$2c\circ(x,y) = (c+c)\circ(x,y) = c\circ(x,y) + c\circ(x,y)$

but

$c\circ(x,y) = (cx,y)$

and

$c\circ(x,y) + c\circ(x,y) = (cx,y) + (cx,y) = (2cx,2y) \ne 2c\circ(x,y)$

Answer 2

As per the comments, check if $(a+b)\mathbf v=a\mathbf v+b\mathbf v$, i.e., "distributivity of scalar multiplication with respect to field addition".

Answer 3

The operation $\oplus$ has $(0,0)$ as neutral element, because $$ (x,y)\oplus(0,0)=(0,0)\oplus(x,y)=(x,y) $$ If we had a vector space, it would be true that $$ 0\circ(x,y)=(0,0) $$ for every $(x,y)$. What about $0\circ(1,1)$?