Vector spaces over the real numbers are familiar creatures. But the definition of a real vector space makes perfect sense when you replace the real numbers ℝ by an arbitrary field F. The crucial thing is that given a non-zero x ∈ F there is a y ∈ F such that xy = 1.

Definition B.0.9 A vector space V over a field F is an abelian group (V, +) with neutral element 0 and a (scalar) multiplication F × V → V denoted (a, v) ↦ av such that

1. (i) (ab)v = a(bv)

2. (ii) 1v = v

3. (iii) (a + b)v = av + bv

4. (iv) a(v + w) = av + aw

for every a, b ∈ F and every v, w ∈ V.

A subspace of V is a subgroup W ⊆ V such that av ∈ W if a ∈ F and v ∈ W. A group homomorphism ϕ: V → W between vector spaces V and W over a field F is called a linear map if ϕ(av) = aϕ(v) where a ∈ F and v ∈ V.

Let ϕ: V → W be a linear map. The subset Ker(ϕ) = {v ∈ V ∣ ϕ(v) = 0} ⊆ V is called the kernel of ϕ and Im(ϕ) = {ϕ(v) ∣ v ∈ V} ⊆ W is called the image of ϕ.