Product of Vector Spaces

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

• Module Product

Proof

This follows directly from Module Product and the definition of vector space.

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 26$: Example $26.5$