Addition and scalar multiplication of linear maps

Weìve already introduced the space of all linear maps ℱ(V, W) of all function f : V → W between two vector spaces V and W over 𝔽 can be made into a vector space over the same field. Vector addition and multiplication of functions are defined as,

where f, g ∈ ℱ(V, W), v ∈ V and α ∈ 𝔽. We are interested in understanding whether this property of linearity is preserved under addition and scalar multiplication of functions.

Theorem 3.4.1 Let f, g : V → W be two linear maps between vector spaces V and W over 𝔽 and α ∈ 𝔽 a scalar. Then the sum f + g and the scalar multiple αf, as defined in Eq. (3.4.1), are linear.

Proof. We need to check the linearity condition (12.1) for f + g and αf, given it is satisfied for f and g.

The proof for αf works analogously and is left as Exercise.  □

We can reformulate this result more abstractly by saying that the set of linear maps f : V → W forms a vector subspace of ℱ(V, W). This vector space of linear maps from V to W is also denoted by Hom(V, W), which stands for homomorphisms from V to W. For the special case V = W, linear maps V → V are also called (vector space) endomorphisms and the vector space of endomorphisms is denoted by End(V).

Theorem 3.4.2 The space Hom(V, W) of linear maps V → W, with addition and scalar multiplication as defined in Eq. 3.4.1, forms a vector spaces over the same field as V and W. If V and W are finite-dimensional its dimension is given by

Proof. It remains to proof the dimension formula. We choose bases (v₁ , ..., v_n) and (w₁ , ..., w_n) of V and W and define the linear maps f_ij ∈ Hom(V, W) for i = 1, ... , n and j = 1, ..., m by

$$f_{ij}({\mathbf{v}}_{\mathbf{k}})=\{\begin{array}{ll}{\mathbf{w}}_{\mathbf{k}}& \text{for }k=i\\ \mathbf{0}& \text{for }k\ne i\end{array}$$

We want to show that these linear maps form a basis of Hom(V, W). For linear independence, start with the equation Σ_ij λ_ij f_ij = 0 and act on the vector v_k which results in Σ_ij λ_kj w_j = 0. Since (w₁, ..., w_n) forms a basis, this implies that all λ_kj = 0. Hence, the f_ij are linearly independent.
Next, consider the function f = Σ_i,j a_ijf_ij, for a_ij ∈ 𝔽. Since f(v_k) = Σ_j a_kj w_j and the w_j are a basis, it follows that any image vectors f (v_k) can be obtained for suitable choices of the a_ij. From Theorem 3.4.1 this means the f_ij span Hom(V, W). Since the number of these functions is nm, Eq. (3.4.2) follows.  □

Map composition and inverse

Linearity is also preserved under map composition and inversion, as the following proposition illustates.

Proposition 3.4.3 Let f, f₁, f₂: V → W and g, g₁, g₂: W → U be linear maps and α₁, α₂ ∈ 𝔽.

1. The composition g ◦ f : V → U is linear.

2. g ◦ (α₁ f₁ + α₂ f₂) = α₁ (g ◦ f₁) + α₂ (g ◦ f₂).

3. (α₁ g₁ + α₂ g₂) ◦ f = α₁ (g₁ ◦ f) + α₂ (g₂ ◦ f).

4. If f⁻¹: W → V exists it is linear.

Poof. (i) All we need to do is check the linearity condition for g ◦ f given it is satisfied for f and g.

(ii) For v ∈ V we have, from the definition of map composition and linearity, that

(iii) This works exactly like the proof for part (ii).

(iv) Let f : V → W be an invertible linear maps with inverse f⁻¹ : W → V. Consider two vectors w₁, w₂ ∈ W. Since f is surjective they can be written as w₁ = f (v₁) and w₂ = f (v₂) for two vectors v₁, v₂ ∈ V, so that v₁ = f⁻¹ (w₁) and v₂ = f⁻¹ (w₂). The linearity condition for f⁻¹ is verified by

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