Linear algebra MOC

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 linalg

dim𝐾⁡𝑉=[𝐿:𝐾]dim𝐿⁡𝑉=(dim𝐾⁡𝐿)(dim𝐿⁡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×𝐽→𝑉(𝑖,𝑗)↦𝛼(𝑖)𝑣(𝑗)

forms an (𝐼 ×𝐽)-indexed 𝐾-basis for 𝑉. Indeed, for any 𝑢 ∈𝑉, we have

𝑢=∑𝑗∈𝐽𝑢𝜆𝑗𝑣(𝑗)

for some finite subset 𝐽𝑢 ⊆𝐽, and for each 𝜆𝑗 ∈𝐿 we have

𝜆𝑗=∑𝑖∈𝐼𝜆𝑗𝜅𝑖𝛼(𝑖)

for some finite subset 𝐼𝜆𝑗 ⊆𝐼. Therefore

𝑢=∑𝑗∈𝐽𝑢∑𝑖∈𝐼𝜆𝑗𝜅𝑖𝛼(𝑖)𝑣(𝑗)

is a finite linear combination.

Corollaries

• Intermediate field extension

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