Vector Spaces problems and axioms I'm currently stuck in two vector space problems. I feel as though the signs and my lack of knowledge throw me off. I am not aware of what $\boxplus, \boxdot$ mean in the first problem (should I just think of them as addition and multiplication). Same happens with problem 2. Should I assume $\oplus, \odot $ mean addition and multiplication. Please help, thank you. 
Problem 1: In ${R}^2$, consider the following operations:
$(x_1, y_1) \boxplus (x_2, y_2) = $(max {$x_1, x_2$}, max { $y_1, y_2$ })
$\alpha \boxdot (x,y) = (\alpha * x, 0)$ 
is ${R}^2$ with these operations a vector space? If your answer is negative, list all the vector spaces axioms that fail to be satisfied and explain why; otherwise prove that all the axioms are satisfied.
Problem 2: In ${R}^2$, consider the following operations:
$(x_1, y_1) \oplus (x_2, y_2) = (x_1 + x_2, 0) 
\alpha \odot (x,y) = (\alpha * x, y)
$
is ${R}^2$ with these operations a vector space? If your answer is negative, list all the vector spaces axioms that fail to be satisfied and explain why; otherwise prove that all the axioms are satisfied. 

The sign $(x_1,y_1)\boxplus (x_2,y_2)$
between the two pairs of numbers defines an new kind of addition 
same with the sign $\boxdot$ in  $\alpha\boxdot(,)=(,0)$a new kind of multiplication by a scalar is defined, you have to find out if with these new definitions in $\mathbb{R}^2$ you have still a Vector space, if not, what axioms do not hold.
