Matrix-matrix multiplication combines transformations, equations, and data into one operation. Each entry of the result comes from one row of the first matrix and one column of the second: multiply matching elements, then add the products. This turns a grid of numbers into a useful geometric or algebraic action. section 2.5.2
Two rules prevent most multiplication mistakes:
- Match inner dimensions: a matrix of width N can multiply only a matrix of height N. section 2.5.2
- Keep the order: A × B generally differs from B × A, and one order may not even be dimensionally valid. section 2.5.2
- Treat vector multiplication as a special case using a thin matrix; in geometry, it transforms a point into another point. section 2.5.2
From equations to a computed vector
An N-dimensional row vector written as a 1 × N matrix can multiply an N × N matrix to yield another 1 × N matrix. Reversing the arrangement gives a matrix-by-column-vector product, the compact form used for a system of linear equations. section 3.5 section 3.5.4
For geometry, multiplication provides a common language for translation, rotation, scaling, and shear. Homogeneous coordinates let these transformations share one matrix-based form, so several operations can be composed into a single transformation matrix. Inverse transformations provide an undo operation. section 4.0 section 4.3.3
Figure 1.2 In 2D, creating a projection is essentially the same as transforming a quadrilateral into some other quadrilateral.
A 2D projection can be viewed as transforming one quadrilateral into another, making the geometric purpose of a transformation concrete. section 1.3
This direct view also matters in production code. A point projection needs only a small amount of arithmetic and can often be expressed as one matrix-multiplication call. Profiling remains essential: wrapper routines and repeated point-representation conversions can cost far more than the arithmetic itself. section 1.3
