Definition:Direct Product of Vector Spaces/Finite Case

Definition
Let $K$ be a field.

Let $V_1,\ldots,V_n$ be $K$-vector spaces.

Let:
 * $\displaystyle V = \prod_{k \mathop = 1}^n V_k$

be their cartesian product.

Let:
 * $+$ be the operation induced on $V$ by the operations $+_1, +_2, \ldots, +_n$ on $V_1, V_2, \ldots, V_n$


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

$\left({V, +, \circ}\right)_k$ is called the direct product of $V_1, \ldots, V_n$.