Linear Transformations
“A matrix is a verb, not a noun—a motion that moves space.”
The formula
T(v) = Av, T(u+v) = T(u)+T(v), T(cu) = c·T(u)How to read it: a map that moves a point v by multiplying it by a matrix A while preserving addition and scaling
- A
- — the matrix defining the map—its columns show where the basis vectors land
- T(u+v)=T(u)+T(v)
- — additivity—move-then-add equals add-then-move
- T(cu)=c·T(u)
- — scaling—stretch-then-move equals move-then-stretch
The hook
A matrix is not a table of numbers—it is a single motion that moves the whole space at once.
In plain words
A linear transformation moves every point of the plane (or space) while keeping grid lines straight, evenly spaced, and the origin fixed. This orderly motion is captured completely by a single matrix.
The intuition
Picture a rubber grid. A linear transformation may stretch, rotate, or shear it, but never bend or tear it. Straight lines stay straight and parallel lines stay parallel. The striking fact: knowing only where the basis vectors (1,0) and (0,1) go fixes the destination of every point—and those two destinations are exactly the two columns of the matrix.
How it's built
Two rules are everything: additivity T(u+v)=T(u)+T(v) and scaling T(cu)=cT(u). Together they let you break any vector into a combination of basis vectors, move each, and add—which is exactly why the matrix-vector product Av is the transformation.
Example
A=[[2,0],[0,3]] stretches by 2 in x and by 3 in y. The point (1,1) goes to (2·1, 3·1)=(2,3). Its columns (2,0) and (0,3) are the destinations of (1,0) and (0,1).
Common mistake
The misconception that matrix multiplication is 'a rule to memorize'. In truth its odd rule is the natural result of applying transformations one after another—which is why order matters (AB ≠ BA).
Where it's used
Rotation, scaling, and projection in computer graphics, coordinate changes for robot arms, dimensionality reduction (PCA) in data science, changing frames in physics—linear transformations are wherever space is reshaped in an orderly way.
Where it came from
Cayley introduced matrices in the mid-1800s as 'tables that record transformations'. A tool for solving equations turned out to be notation for motions of space.
Prerequisites
Quick check
Apply the matrix [[0,−1],[1,0]] to the vector (1,0). (This matrix is a 90° rotation.)
- (1,0)
- (0,1)✓
- (−1,0)
- (0,−1)
Practice
Apply A=[[2,0],[0,2]] to (3,5); find the x-component of the image.
Answer: 6
- x-component = 2·3 + 0·5
- = 6
Takeaway: A diagonal matrix stretches each axis by its entry.
Apply the shear A=[[1,1],[0,1]] to (2,3); find the x-component of the image.
Answer: 5
- x-component = 1·2 + 1·3
- = 5 (the y-component stays 3)
Takeaway: A shear slides sideways in proportion to height.
Apply A=[[3,0],[0,3]] to (4,4); find the y-component of the image.
Answer: 12
- y-component = 0·4 + 3·4
- = 12
Takeaway: The scale-by-3 matrix pushes every point 3× from the origin.
Explain in your notebook why knowing where (1,0) and (0,1) go determines where every point goes.
Answer: undefined
- any point (a,b) = a·(1,0) + b·(0,1)
- by additivity and scaling, T(a,b) = a·T(1,0) + b·T(0,1)
- so the two basis destinations decide everything
Takeaway: Linearity = know the parts (basis) and the whole follows.