5.3 Linear maps

We’ve looked at linear functions several times now to get a concrete idea of how they work. We must generalize this idea for vector spaces.

Let U and V be vector spaces over the same field F. The function L: U → V is a linear map

• if u₁ and u₂ are in U then L(u₁ + u₂) = L(u₁) + L(u₂), and
• if a₁ is in F then L(a₁ u₁) = a₁ L(u₁).

In particular, for a₂ also in F we have

L(a₁ u₁ + a₂ u₂) = a₁ L(u₁) + a₂ L(u₂).

When U = V we also say L is a linear transformation of U or a linear operator on U.

All linear transformations on R² look like

(x, y) ↦ (ax + by, cx + dy)

using Cartesian coordinates and a, b, c, d, x, and y in R. This is interesting because the linear transformations on R¹ all look like the somewhat trivial (x) ↦ (ax). To take it further, the linear transformations on R³ look like the messy

(x,y,z) ↦ (ax + by...