I am trying to determine which axioms of a vector space do not hold for the following:
Let $V$ be $\mathbb{R}^2$, but with scalar multiplication defined as:
$a(x_1, y_1) = (ax_1, y_1)$
I can't find which axiom fails.
However, I don't think this is a vector space as I know, if $V$, with $u \in V$ is a vector space then $(-1)u = -u$, but this is not true for $V$.
What have I missed?
Best Answer
I agree with Gary's comment. Excellent way to show that under this product $\mathbb R^2$ is not a vector space.
Here is an alternative way, I am assuming that the sum of two vectors in $\mathbb R^2$ is defined as usual sense i.e. $(x,y)+(u,v)=(x+u,y+v)$, and the scalar product is defined as $c(x,y)=(cx,y)$. Then the following contradiction occurs, $$(1,1)+(1,1)=(2,2)\ne (2,1)=2\times(1,1).$$