Let $V$ be the set of all ordered pairs of real numbers with addition defined by $$(x_1,x_2)+(y_1,y_2)=(x_1+y_1,x_2+y_2)$$ and scalar multiplication defined by $$\alpha\circ(x_1,x_2)=(\alpha x_1,x_2)$$ Is $V$ a vector space with these operations?
$\begingroup$
$\endgroup$
Add a comment
|
$\begingroup$
$\endgroup$
2
If $V$ is a vector space then, for any $v\in V$, $0v=(0+0)v=0v+0v\implies 0v=0$.
Your definition of scalar multiplication is that $a(x_1,x_2)=(ax_1,x_2)$.
Therefore, $0(1,2)=(0,2)\ne (0,0)$ so $V$ is not a vector space.
edm pointed out that I have not shown that $(0,0)$ is the zero vector. For any $(x,y)\in V$, $(x,y)+(0,0)=(0,0)+(x,y)=(x,y)$. Therefore, $(0,0)$ is the identity element and hence, the zero vector.
-
$\begingroup$ Be extra careful. You have not yet proved that $(0,0)$ is the zero vector. $\endgroup$ – edm Oct 30 '18 at 4:15
-
$\begingroup$ @edm I added that. Thanks for pointing it out. $\endgroup$ – John Douma Oct 30 '18 at 4:19
