Let $V$ be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operations on $u$ and $v$

Question

I'm confused on b) and c) , what do the terms in it mean? I'm also having trouble finding e) ..

Answer 1

Just check two things:

1) $(x,y) + (0,0) \neq (x,y)$

2) $(x,y) + (-1,-1) = (x,y)$

Stop thinking of $\mathbf{0}$ as the real number. This is just a symbol that denotes the additive identity. And that could be anything, depending of the operations you define in your set. Some texts like to use $\oplus$ and $\odot$ instead of $+$ and $\cdot$, to remind you that these operations are not the usual ones. The concepts of $\mathbf{0}$ and $\mathbf{1}$ depend strongly of the operations defined in the set.

Answer 2

For e, you might notice that the vector addition law is not the usual one, but the scalar multiplication law is. Two of the axioms combine these two operations. To show they don't hold, you just need to find one example for each, so try some examples.