Definition:Direct Product of Vector Spaces/Finite Case

Definition

Let $K$ be a field.

Let $V_1, V_2, \ldots, V_n$ be $K$-vector spaces.

Let:

$\ds V = \prod_{k \mathop = 1}^n V_k$

be their cartesian product.

Let:

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

$\circ$ be defined as $\lambda \circ \tuple {x_1, x_2, \ldots, x_n} := \tuple {\lambda \circ x_1, \lambda \circ x_2, \ldots, \lambda \circ x_n}$

$\struct {V, +, \circ}_K$ is called the direct product of $V_1, \ldots, V_n$.

Also see

• Direct Product of Vector Spaces is Vector Space

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.5$