Vector Space

Definition : A vector space is a nonempty set V with two operation (addition, scalar, multiplication ) that satisfy the following conditions:

\begin{array}{l} 1) u + v b e l o n g s V w h e n e v e r u, v b e l o n g s V \\ 2) u + v = v + u f o r a l l u, v b e l o n g s V \\ 3) (u + v) + w = u + (v + w) f o r a l l u, v, w b e l o n g s V \\ 4) T h e r e e x i s t s 0_{v} b e l o n g s V s u c h t h a t 0_{v} + u = u \\ 5) f o r a l l u b e l o n g s V, t h e r e e x i s t s (- u) b e l o n g s V s u c h t h a t u + (- u) = 0_{v} \\ 6) k u b e l o n g s V f o r a l l k b e l o n g s F a n d u b e l o n g s V (F = r e a l s O r c o m p l e x) \\ 7) k (u + v) = k u + k v, k b e l o n g s F, u, v b e l o n g s V . \\ 8) (k_{1} + k_{2}) u = k_{1} u + k_{2} u, k_{1}, k_{2} b e l o n g s F, u b e l o n g s V \\ 9) k_{1} (k_{2} u) = (k_{1} k_{2}) u \\ 10) 1 u = u \end{array}