Linear Transformations

Walkthrough

Verify that T(x, y) = (2x, x + y) is a linear transformation.

1. 1.A transformation T is linear if it satisfies two properties: (1) T(u + v) = T(u) + T(v), and (2) T(cu) = cT(u).
2. 2.Check additivity: T(x₁ + x₂, y₁ + y₂) = (2(x₁+x₂), (x₁+x₂) + (y₁+y₂)) = (2x₁+2x₂, x₁+x₂+y₁+y₂).
3. 3.T(x₁,y₁) + T(x₂,y₂) = (2x₁, x₁+y₁) + (2x₂, x₂+y₂) = (2x₁+2x₂, x₁+y₁+x₂+y₂). These are equal. ✓
4. 4.Check homogeneity: T(cx, cy) = (2cx, cx + cy) = c(2x, x + y) = cT(x, y). ✓
5. 5.Both properties hold, so T is a linear transformation.