Question

"let v = r 2 with the usual addition and scalar multiplication defined by k(u1, u2) = (ku1, 0). determine which of the five axioms of vector spaces involving scalar multiplication v satisfies and which fail. for the ones it satisfies, prove that it satisfies the axiom. for those that fail, show that it fails with a counterexample."

Answer 1

Let $\mathbf u\in\mathbb R^2$, where

$\mathbf u=(u_1,u_2)$

and let $k\in\mathbb R$ be any real constant.

Given this definition of scalar multiplication, we can see right away that there is no identity element $e$ such that

$e\mathbf u=\mathbf u$

because

$e\mathbf u=e(u_1,u_2)=(eu_1,0)\neq(u_1,u_2)=\mathbf u$