Geometric Bites

Linear Transformations in ℝ³ and 3×3 Matrices Matrices give us a compact way to describe linear transformations in three-dimensional space. A linear transformation is a mapping T : ℝ³ → ℝ³ that sends a point with position vector (x, y, z) to another point, according to a rule with two key properties. What Makes a Transformation Linear? A transformation T : ℝ³ → ℝ³ is called linear if, for all real numbers λ and all vectors (x, y, z) in ℝ³, T(λx, λy, λz) = λ T(x, y, z), and for all vectors (x₁, y₁, z₁) and (x₂, y₂, z₂) in ℝ³, T(x₁ + x₂, y₁ + y₂, z₁ + z₂) = T(x₁, y₁, z₁) + T(x₂, y₂, z₂). The point that (x, y, z) is sent to is called the image of (x, y, z) under T. The Standard Basis Vectors To find the matrix that represents a particular transformation, it is enough to know what happens to three special vectors, called the standard basis for ℝ³: î = (1, 0, 0) ĵ = (0, 1, 0) k̂ = (0, 0, 1) Once we know the images of î, ĵ and k̂, th...

Understanding Eigenvectors and Eigenvalues: A Geometric Perspective Every linear transformation has a hidden structure. Most vectors are pushed into new directions when a matrix acts on them, but a handful of special vectors behave differently. These are the eigenvectors — directions that remain perfectly aligned with themselves, even after the transformation has been applied. To understand how a matrix works, you must understand these special directions. What Is an Eigenvector? An eigenvector of a matrix A is a non-zero vector x that satisfies the relation A x = λ x The number λ is the eigenvalue associated with x . This equation expresses a simple but striking fact: the transformation does not rotate the vector at all. The direction is preserved exactly. The only change is a scaling by the factor λ . A positive eigenvalue stretches the vector. A value between 0 and 1 compresses it. A negative eigenvalue reverses the direction. But in every case, the vector re...

Homogeneous Coordinates: A Simple and Intuitive Primer In ordinary geometry, we use familiar coordinates such as (x, y) in 2D or (x, y, z) in 3D. These work well, but they have one major limitation: not all geometric transformations fit neatly into this system—especially translations and perspective projections. To unify everything into one clean mathematical framework, we introduce homogeneous coordinates . They provide a simple way to treat every transformation—from translations to perspective projection— using only matrix multiplication. 1. Why Do We Need Something New? In ordinary coordinates: rotations are matrices, scalings are matrices, shears are matrices, translations are not matrices . Translation is the “odd one out.” This creates friction in computer graphics, robotics, and projective geometry, where we want one system that handles everything the same way. Homogeneous coordinates fix this by adding one extra coordinate. 2. The Bas...