I'm trying to prove an axiom of a vector space - namely, distributivity of scalar multiplication over addition. In the answer section of the book, it says that the property holds. When I do it, it doesn't seem that it does. This is what I have:
In order for a set to be a vector space it has to satisfy,
$$\forall \vec x, \vec y \in V: \forall c \in \Bbb R,\ c(\vec x + \vec y) = c \vec x + c \vec y $$
For my problem, vector addition has been defined as, $$(x_1,x_2)\ +' \ (y_1, y_2) = \ (x_1 + 2y_1, 3x_2 - y_2)$$
What I Have
Let $c$ be a scalar s.t. $c \in \Bbb R$. Then, $$c(\vec x \ +' \vec y) = c((x_1, x_2) \ +' \ (y_1, y_2)) = c(x_1 +2y_1, 3x_2 - y_2) = (c(x_1 + 2y_1) \ + \ c(3x_2 - y_2)) \\ = (cx_1 + c2y_1, c3x_2 - cy_2) = \ (cx_1, c3x_2) \ +' \ (c2y_1, cy_2) = c(x_1, 3x_2) \ +' \ c(2y_1, y_2)$$
Which is not equal to $c \vec x + c \vec y$. So it seems the property doesn't hold for this definition of vector addition. What am I doing wrong? Thanks in advance.