Spans and basis
What you’ll learn
Section titled “What you’ll learn”This is lesson 2 of Track 4 (Visual Math: Linear Algebra). The previous lesson left you with exactly two operations, adding vectors and scaling them, and this lesson is where that small toolkit starts paying off. By the end you will be able to look at a set of vectors and predict what they can reach: the whole plane, just a line, or only the origin, and say why. Along the way you will pin down three of the most-used words in the subject, span (everything reachable), basis (the smallest set that reaches everything), and dimension (the size of that set), plus the idea of linear independence that ties them together. The source curriculum is 3Blue1Brown’s Essence of Linear Algebra by Grant Sanderson, freely available at 3blue1brown.com.
Where this fits
Section titled “Where this fits”This is lesson 2 of 15, the second step of Phase 1 (geometric foundations). It builds directly on the two operations from What vectors actually are: a linear combination is just those two operations used together. The next lesson, Linear transformations as moves, turns these still pictures into motion: once you know what a basis is, the natural question is what happens to the whole space when you move the basis vectors, and that single idea defines every linear transformation.
Before you start
Section titled “Before you start”Prerequisites: the previous lesson, What vectors actually are. You need to be comfortable with adding two vectors component by component and scaling a vector by a number, because span is built entirely out of those two moves. No other background is needed, and the practice is pen and paper.
About the math
Section titled “About the math”The new work here is light. You will add and scale vectors (which you already do), and in one practice exercise you will solve a small two-equation system by hand (two unknowns, no calculator) to find a vector’s coordinates in a non-standard basis. That single worked system is the most algebra the lesson asks for; everything else is the geometric picture of reaching points by adding and scaling.
By the end, you’ll be able to
Section titled “By the end, you’ll be able to”- Define a linear combination and describe the span of a set of vectors as everything reachable by adding and scaling
- Classify the span of two vectors in the plane as the whole plane, a line, or the origin, based on whether they are independent
- Explain what a basis is (linearly independent and spanning) and why coordinates are the amounts of basis vectors
- Distinguish linearly dependent from independent sets and explain what dependence costs (a dimension of reach)
- Relate the dimension of a space to the size of a basis for it
Time and difficulty
Section titled “Time and difficulty”- Read time: about 9 minutes
- Practice time: about 15 minutes (a classify-the-span drill, a coordinates-in-a-new-basis exercise, and flashcards)
- Difficulty: intro (a foundational lesson; the only algebra is one small two-equation system solved by hand)