Abstract Linear Algebra 23 | Combinations of Linear Maps

Linear maps aren’t just single functions between vector spaces—they form their own vector space under addition and scalar multiplication. Given two linear maps K, L: V → W (with the same domain V and codomain W over the same field), their sum is defined pointwise by (K + L)(x) = K(x) + L(x), and scalar multiples are defined by (λL)(x) = λ·L(x). With these operations, the collection Hom(V, W) of all linear maps from V to W becomes an F-vector space again: the zero element is the zero map sending every x ∈ V to 0 ∈ W, and the usual vector-space rules (closure, associativity, distributivity, etc.) carry over because they hold in W for the outputs.

This abstract “calculus of linear maps” becomes concrete through orthogonal projections. For a finite-dimensional inner product space V with an orthonormal basis {e1, e2, …}, take a subspace U spanned by n − 1 of those basis vectors. The orthogonal projection onto U is a linear map P_U: V → V given by P_U(x) = Σ_{j=1}^{n−1} e_j · ⟨e_j, x⟩, using the inner product’s linearity in the second argument. The complementary projection onto U⊥ is also linear and can be written as a single-term sum involving the remaining basis vector (the “normal component”). Adding the two projections recovers the identity map on V: P_U + P_{U⊥} = Id_V. Subtracting them yields a reflection across U, expressible as Id_V − 2·P_{U⊥} (or equivalently Id_V − 2 times the normal component projection), which still remains linear.

Beyond combining linear maps by addition and scaling, linear maps also combine by composition. If K: U → V and L: V → W are linear, then the composition L ∘ K: U → W is linear as well. This matters because composition behaves like a structural operation on Hom(V, W), analogous to multiplication in familiar settings like matrices.

The projection example shows composition’s effect sharply. Composing a projection with itself gives the same projection again: projecting twice changes nothing. Composing the two orthogonal projections (onto U and onto U⊥) collapses everything to zero: since the subspaces are orthogonal, applying one projection after the other always produces the zero vector, so the resulting linear map is the zero map in Hom(V, V). Taken together, these results show that linear maps support both vector-space operations (addition/scalar multiplication) and an additional “multiplication-like” operation (composition), setting up a richer algebraic structure that will be developed further later through matrix multiplication on a concrete level.