Definition:Vector Space on Cartesian Product
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Theorem
Let $\struct {K, +_K, \times_K}$ be a division ring.
Let $n \in \N_{>0}$.
Let $+: K^n \times K^n \to K^n$ be defined as:
- $\tuple {\alpha_1, \ldots, \alpha_n} + \tuple {\beta_1, \ldots, \beta_n} = \tuple {\alpha_1 +_K \beta_1, \ldots, \alpha_n +_K \beta_n}$
Let $\times: K \times K^n \to K^n$ be defined as:
- $\lambda \times \tuple {\alpha_1, \ldots, \alpha_n} = \tuple {\lambda \times_K \alpha_1, \ldots, \lambda \times_K \alpha_n}$
Then $\struct {K^n, +, \times}_K$ is the $K$-vector space $K^n$.
Also see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.1$
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 32$. Definition of a Vector Space: Example $62$