What vectors actually are
What you’ll learn
Section titled “What you’ll learn”This is the opening lesson of Track 4 (Visual Math: Linear Algebra), and it sets the foundation the whole track stands on. By the end you will be able to move fluently between the three views of a vector (the arrow you picture, the list of numbers you compute with, and the abstract “anything you can add and scale” view) and pick the right frame for the problem in front of you. Along the way you will see how a coordinate system translates exactly between arrows and lists, and you will pin down the two operations, addition and scalar multiplication, that actually define a vector and that every later topic in the track is built from. 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 1 of 15 and the entry point of the track, so there is nothing before it. Its job is to make the most basic object solid before any machinery arrives, because everything later (spans, linear transformations, the determinant, eigenvectors) is assembled out of the two operations introduced here. The next lesson, Spans and basis, asks the natural follow-up: starting from a few vectors, which points can you reach by adding and scaling them? That question, and the rest of the track, rests directly on this lesson.
Before you start
Section titled “Before you start”Prerequisites: none. You do not need any prior linear algebra, and no software is required; the practice is pen and grid paper. Comfort with the idea of an array or list of numbers (the kind you meet in any programming language) helps the computer science view land faster, but it is not required.
About the math
Section titled “About the math”There is almost no math to fear here. The only arithmetic is adding two short lists of numbers and multiplying a list by a single number, both of which you can do in your head or on paper. The lesson leans on pictures (arrows on a grid) far more than on symbols, and the practice is a hands-on, no-calculator exercise. If notation has been the thing that blocked you before, this lesson is built to get you past that wall, not into it.
By the end, you’ll be able to
Section titled “By the end, you’ll be able to”- Distinguish the arrow, list, and abstract (add-and-scale) views of a vector and pick the right frame for a given problem
- Translate a vector between its arrow form and its coordinate-list form using a coordinate system
- Add two vectors component by component and predict the result geometrically (tip to tail)
- Scale a vector by a number and describe the geometric effect (stretch, squish, or flip)
- Explain why a vector’s coordinates are a description in a chosen frame, not the vector itself
Time and difficulty
Section titled “Time and difficulty”- Read time: about 10 minutes
- Practice time: about 15 minutes (a move-between-the-views exercise on grid paper, a pick-the-right-frame drill, and flashcards)
- Difficulty: intro (a foundational lesson; only the lightest arithmetic, no prior math needed)