Direct Product of Vector Spaces is Vector Space

Theorem
Let $G_1, G_2, \ldots, G_n$ be $K$-vector spaces.

Let:
 * $\displaystyle G = \prod_{k \mathop = 1}^n G_k$

be the cartesian product of $G_1, G_2, \ldots, G_n$.

Then $\left({G, +, \circ}\right)_K$ is a $K$-vector space where:


 * $+$ is the operation induced on $G$ by the operations $+_1, +_2, \ldots, +_n$ on $G_1, G_2, \ldots, G_n$


 * $\circ$ is defined as $\lambda \circ \left({x_1, x_2, \ldots, x_n}\right) := \left({\lambda \circ x_1, \lambda \circ x_2, \ldots, \lambda \circ x_n}\right)$

Also see

 * Definition:Module Direct Product

Proof
This follows directly from Finite Direct Product of Modules is Module and the definition of vector space.