# Determine which of the following are Vector Spaces

Going off the axioms, I'm trying to determine how to verify each of these. In the first set, do I take $$u = (x_1, x_2, x_3)$$ and $$v = (y_1, y_2, y_3)$$, or would it rather be $$u = (x_1 + y_1, x_2 + y_2, x_3 + y_3), v = (x_4 + y_4, x_5 + y_5, x_6 + y_6)$$?

Based on what I know about scalar multiplication, I would say the first set is not a vector space because $$0(x_1, x_2, x_3)$$ = $$(0, 0, 0)$$ which is not equivalent to $$(0x_1, x_2, x_3)$$, for non-zero values of $$x_2$$ and $$x_3$$. The second and third sets seem to define vector spaces, as I can't seem to figure out any axiom that directly fails (i.e. $$u+v$$ is in $$V$$, a unique additive identity exists, and the set is closed under scalar multiplication). Any help would be appreciated.

Edit: The third set is not a vector space.

The third one is not a vector space. The zero vector in this space is $$(-1,-1)$$ but if you multiply this by $$2$$ you don't get back the zero vector.
• Ahh thank you! My Linear Algebra is very rusty right now, so these problems are helping me refresh. For the first set, what would you set the $u$ and $v$ vectors as when checking for $u+v$ Jan 22 '19 at 5:46
• Right, I figured as much, but to verify that axiom of addition holds, do I let u = $(x_1, x_2, x_3)$ or rather u = $(x_1 + y_1,...)$ Jan 22 '19 at 5:52