AI Tutor
Every time a video game rotates a character or scales a 3D model, it is silently multiplying thousands of coordinate points by a single 2×22 \times 2 grid of numbers.

Points as Column Vectors

In the Cartesian plane, any point (x,y)(x, y) can be represented as a column vector (xy)\begin{pmatrix} x \\ y \end{pmatrix}, which points from the origin (0,0)(0,0) directly to that location.
A 2×22 \times 2 transformation matrix is a two-row by two-column array of numbers that acts like a function, taking in an input vector and spitting out a transformed output vector.
How does a matrix actually change those numbers? Let's look at the multiplication rule.

Matrix-Vector Multiplication

In the 1850s, British mathematician Arthur Cayley developed matrix algebra to compactly describe systems of linear equations. To multiply a matrix by a vector, you take the dot product of each matrix row with the column vector.
Notice the structure: the top row gives the new x′x'-coordinate (ax+byax + by), and the bottom row gives the new y′y'-coordinate (cx+dycx + dy).
What does this look like with real coordinates on a grid?

Worked Example: Mapping a Point

Let's transform the point P(3,1)P(3, 1) using the matrix A=(2−112)A = \begin{pmatrix} 2 & -1 \\ 1 & 2 \end{pmatrix}.
We multiply row-by-column: x′=(2)(3)+(−1)(1)=6−1=5x' = (2)(3) + (-1)(1) = 6 - 1 = 5, and y′=(1)(3)+(2)(1)=3+2=5y' = (1)(3) + (2)(1) = 3 + 2 = 5, giving the transformed point P′(5,5)P'(5, 5).