Linear Transformations — Matrices as Moving Space
A matrix isn't a static block of numbers. It's an instruction for moving every point in space at once — and it's completely determined by where it sends two arrows.
Intuition First
Instead of picturing a matrix as a grid of numbers, picture it as an action: something that grabs every point in space and moves it somewhere new — stretching, rotating, or shearing the whole plane at once.
A transformation is called linear if it plays fair: the origin never moves, straight lines stay straight, and evenly spaced grid lines stay evenly spaced everywhere. Once you know where a linear transformation sends just the two basis arrows, i = (1,0) and j = (0,1), you already know where it sends every other point — because every point is some recipe of i and j, the linear combination idea from the last lesson.
Start With a Number You Can Check by Hand
Take the matrix that sends i to (2, 0) and j to (1, 2). Apply it to the point (1, 1):
Each new coordinate is just a linear combination of the old ones. The sliders below start on this exact matrix.
Try It — Warp the Grid
Move the sliders to change the matrix. Watch the grid stretch and shear, and watch where the test point (1, 1) lands.
Formal Definition
A transformation T is linear if it satisfies two properties for all vectors u, v and scalars c:
Every linear transformation of the plane can be written as multiplication by a matrix, whose columns are exactly where the basis vectors land: