Linear transformations as moves

What you’ll learn

This is lesson 3 of Track 4 (Visual Math: Linear Algebra), and it is where the matrix finally stops being a mystery grid. By the end you will be able to look at any 2x2 matrix and sketch what it does to space, without computing a single interior point, because you will know the one fact that explains every matrix rule: a matrix is just a record of where the two basis vectors land, written as columns. You will see why following only i-hat and j-hat is enough to know the fate of every vector, and why matrix-vector multiplication is the linear-combination idea from the previous lesson rather than an arbitrary procedure. The source curriculum is 3Blue1Brown’s Essence of Linear Algebra by Grant Sanderson, freely available at 3blue1brown.com.

Where this fits

This is lesson 3 of 15, the third step of Phase 1 (geometric foundations). It uses the basis idea from Spans and basis directly: because every vector is a combination of i-hat and j-hat, tracking those two is enough to track them all. The next lesson, Matrix multiplication as composition, asks the natural follow-up: if one matrix is one move, what does it mean to do two moves in a row? The answer turns matrix multiplication into “do this, then that,” and shows it is not an arbitrary rule either.

Before you start

Prerequisites: the previous lesson, Spans and basis. You need to be comfortable with the idea that every vector is a linear combination of the basis vectors i-hat and j-hat, because the whole lesson rests on a transformation preserving that combination. The two operations from lesson 1 (adding and scaling) are also assumed. The practice is pen and paper.

About the math

The arithmetic is light and concrete: you will write a few 2x2 matrices by deciding where the basis vectors should land, and apply each to a test vector using the same scale-and-add you already know. There are no formulas to memorize; matrix-vector multiplication is built up from the linear combination you met in the span lesson. The emphasis is geometric throughout, reading a transformation off the picture of where the basis went.

By the end, you’ll be able to

• State the two requirements that make a transformation linear (origin fixed, lines stay straight and evenly spaced)
• Explain why knowing where i-hat and j-hat land determines the transformation of every vector
• Read a 2x2 matrix as the two transformed basis vectors written as columns, and compute M·v as a linear combination of those columns
• Sketch what a 2x2 matrix does to the unit square by plotting its columns and drawing the parallelogram they span
• Distinguish a linear transformation from an affine one (a fixed shift of the whole plane)

Time and difficulty

• Read time: about 10 minutes
• Practice time: about 15 minutes (a build-the-matrix-and-apply-it exercise, a sketch-the-unit-square drill, and flashcards)
• Difficulty: intro (foundational; the only arithmetic is scale-and-add applied to a matrix’s columns)