gen-define-a-linear-map-and-give-one-short-worked-example-0a3b

Define a linear map and give one short worked example.

# Linear Map ## Definition A **linear map** (or linear transformation) is a function T: V → W between two vector spaces V and W that preserves vector addition and scalar multiplication: 1. **T(u + v) = T(u) + T(v)** for all u, v ∈ V 2. **T(cv) = cT(v)** for all v ∈ V and scalars c ## Worked Example Let T: ℝ² → ℝ² be defined by T(x, y) = (2x + y, 3y). **Verify it's linear:** Check property 1: - T((x₁, y₁) + (x₂, y₂)) = T(x₁ + x₂, y₁ + y₂) = (2(x₁ + x₂) + (y₁ + y₂), 3(y₁ + y₂)) = (2x₁ + y₁ + 2x₂ + y₂, 3y₁ + 3y₂) = (2x₁ + y₁, 3y₁) + (2x₂ + y₂, 3y₂) = T(x₁, y₁) + T(x₂, y₂) ✓ Check property 2: - T(c(x, y)) = T(cx, cy) = (2cx + cy, 3cy) = c(2x + y, 3y) = cT(x, y) ✓ **Example calculation:** T(1, 2) = (2(1) + 2, 3(2)) = **(4, 6)**