Maths 3.2Lesson outlineLesson 2 of 7 in Module 3
Matrix Multiplication as Transformation
Read a matrix as something that moves space: rotate, stretch, shear, project.
Video in production.
Learn
In this lesson11 parts
- 01A matrix that takes in a point
- 02Where the basis vectors land
- 03Rotate, stretch, shear and project
- 04The determinant
- 05One transformation after another
- 06Why order matters
- 07The whole grid in one product
- 08Checkpoint: transformations so far
- 09Standardising the ward as a transformation
- 10Transformations in NumPy
- 11Recap, challenge, and next: the inverse
Key terms
The words this lesson introduces, each in one line. The module’s glossary collects them all.
- basis vectors
e₁ = (1, 0)ande₂ = (0, 1); every point is a combination of them |(1, 2) = e₁ + 2e₂- columns of a matrix
- where
e₁ande₂land |[[2, 1], [1, 3]]:(2, 1)and(1, 3) - linear transformation
- moves the plane so grid lines stay straight, parallel and evenly spaced, with the origin fixed | every matrix
- projection
- a transformation that flattens the plane onto a line |
[[1, 0], [0, 0]] - determinant
- the factor a matrix scales every area by;
ad − bcfor a 2 × 2 matrix |5 - sign of the determinant
- negative when the transformation flips the plane over | the flip,
−1 - composition
- one transformation applied after another | shear, then rotate
- matrix product
- the single matrix that does both |
RH = [[0, −1], [1, 1]] - shape rule for products
(m, k) @ (k, n) → (m, n)|(2, 3) @ (3, 4) → (2, 4)
Quiz
Your first pick on each question is the one that counts, and a right one earns a coin. Getting one wrong here is how the lesson sticks.