Linear transformations

What you'll learn

1. 1

   Imagine a rubber sheet stretched on a frame. A linear transformation is a way to stretch, squish, rotate or shear that sheet — but the grid lines stay straight and evenly spaced!

2. 2

   Here's a 2D grid with a blue arrow from (0,0) to (1,0) and a red arrow to (0,1). A linear transformation maps these to new positions.

3. 3

   Let's see how the transformation matrix [[2, 0], [0, 3]] works on the point (1, 1).

4. 4

   Drag the point (1, 0) to where it goes after applying the transformation matrix [[1, 2], [0, 1]] (a shear).

5. 5

   What happens to the point (1,0) under the matrix [[0, -1], [1, 0]]?

6. 6

   Now apply the matrix [[1, 1], [0, 1]] (a shear) to the point (0,2) yourself. Type the new coordinates.

7. 7

   Which of these is NOT a linear transformation? (Hint: does it keep the origin fixed and grid lines parallel?)