Vector space over a field extension

Let $𝐿 : 𝐾$ be a field extension and $𝑉$ be an $𝐿$ -vector space. Then $𝑉$ is also a $𝐾$ -vector space, and #m/thm/linalg

d i m 𝐾 𝑉 = [𝐿 : 𝐾] d i m 𝐿 𝑉 = (d i m 𝐾 𝐿) (d i m 𝐿 𝑉)

Proof

That $𝑉$ is a vector space over any subfield of $𝐿$ , so in particular it is an $𝐿$ -vector space. Let $𝛼 : 𝐼 ↪ 𝐿$ be an $𝐼$ -indexed $𝐾$ -basis for $𝐿$ and $𝑣 : 𝐽 ↪ 𝑉$ be a $𝐽$ -indexed $𝐿$ -basis for $𝑉$ . We claim that

𝛼 : 𝐼 \times 𝐽 \to 𝑉 (𝑖, 𝑗) \mapsto 𝛼 (𝑖) 𝑣 (𝑗)

forms an $(𝐼 \times 𝐽)$ -indexed $𝐾$ -basis for $𝑉$ . Indeed, for any $𝑢 \in 𝑉$ , we have

𝑢 = \sum 𝑗 \in 𝐽 𝑢 𝜆 𝑗 𝑣 (𝑗)

for some finite subset $𝐽 𝑢 \subseteq 𝐽$ , and for each $𝜆 𝑗 \in 𝐿$ we have

𝜆 𝑗 = \sum 𝑖 \in 𝐼 𝜆 𝑗 𝜅 𝑖 𝛼 (𝑖)

for some finite subset $𝐼 𝜆 𝑗 \subseteq 𝐼$ . Therefore

𝑢 = \sum 𝑗 \in 𝐽 𝑢 \sum 𝑖 \in 𝐼 𝜆 𝑗 𝜅 𝑖 𝛼 (𝑖) 𝑣 (𝑗)

is a finite linear combination.

Corollaries

Intermediate field extension

#state/tidy | #lang/en | #SemBr