Stepping up to 3D

What you’ll learn

This is lesson 5 of Track 4 (Visual Math: Linear Algebra), and it closes Phase 1 by stepping off the flat page into the space you actually live in. By the end you will be able to predict how a 3D transformation moves the unit cube, by plotting where the three basis vectors land and drawing the slanted box (a parallelepiped) they span. The deeper point is reassurance: the jump to 3D introduces no new rules. You add one basis vector, the matrix gains a column, the product gains a term, and everything from the previous four lessons runs untouched. That same jump is exactly the one that scales to the hundreds of dimensions a real model uses. The source curriculum is 3Blue1Brown’s Essence of Linear Algebra by Grant Sanderson, freely available at 3blue1brown.com.

Where this fits

This is lesson 5 of 15, the final lesson of Phase 1 (geometric foundations). It lifts the 2D machinery of lessons 1 to 4 straight into 3D, so it is mostly consolidation rather than new theory. It also closes the on-ramp: after this, Phase 2 (geometry of operations) starts asking sharper questions about what a transformation does. The next lesson, The determinant, asks by how much a transformation stretches or squashes the space it acts on, a single number you can read off the cube (or the square) you just learned to sketch.

Before you start

Prerequisites: the previous lesson, Matrix multiplication as composition, and ideally all of Phase 1. You need to be comfortable reading a matrix as the transformed basis vectors written as columns and computing M · v as a linear combination of those columns. This lesson adds a third basis vector and a third column to ideas you already have; it does not introduce new machinery. The practice is pen and paper.

About the math

The arithmetic is the same scale-and-add, now with a third term. You will apply a few 3x3 matrices (rotations about an axis, a diagonal scaling) to a test vector, and read what the unit cube becomes from a matrix’s three columns. No new formulas appear; the lesson’s whole claim is that 3D adds a dimension, not a difficulty.

By the end, you’ll be able to

• Extend the definition of a linear transformation to 3D (origin fixed, grid lines straight and even in three directions)
• Name the three standard basis vectors and write a 3D vector as a combination of them
• Read a 3x3 matrix as three transformed basis vectors and compute M·v as x·col1 + y·col2 + z·col3
• Predict how a 3D transformation moves the unit cube by plotting the three columns as the edges of a parallelepiped
• Explain why the 2D-to-3D leap is the same leap to any number of dimensions

Time and difficulty

• Read time: about 10 minutes
• Practice time: about 15 minutes (an apply-the-3x3-matrix exercise, a read-the-unit-cube drill, and flashcards)
• Difficulty: intro (foundational and largely consolidation; the arithmetic is the previous lessons’ product with one more term)