Vectors, what even are they?
Essence of Linear Algebra ยท Chapter 1 ๐ https://youtu.be/fNk_zzaMoSs
๐ง Big Idea
Linear algebra is usually taught too numerically. The real power comes from geometric intuition - seeing vectors as arrows in space, not just lists of numbers. Both views are valid and you need to move fluidly between them.
๐ What Is a Vector? (3 perspectives)
| Perspective | View of a vector |
|---|---|
| Physics | An arrow with a length and a direction โ can be moved around freely |
| Computer Science | An ordered list of numbers (e.g. house size + price = 2D vector) |
| Mathematics | An element of a vector space โ anything obeying the 10 axioms of addition & scalar multiplication |
3b1b's synthesis: Think of a vector as an arrow rooted at the origin of a coordinate system. The tip's position is what the numbers describe.
๐บ๏ธ Vectors in a Coordinate System
- 2D: A pair
[x, y]- x โ how far to move right/left along the x-axis
- y โ how far to move up/down parallel to the y-axis
- 3D: A triplet
[x, y, z]โ z-axis is perpendicular to both
The coordinates are instructions from the origin to the tip of the arrow.
โ Vector Addition
Geometrically:
- Place the tail of vector B at the tip of vector A
- Draw a new arrow from the tail of A to the tip of B โ that's A + B
Intuition: Each vector is a "step". Adding them = taking both steps in sequence. The result is the single step that gets you to the same place.
Numerically:
[xโ, yโ] + [xโ, yโ] = [xโ+xโ, yโ+yโ]
Each component adds independently.
โ๏ธ Scalar Multiplication
Multiplying a vector by a plain number (a scalar) scales it:
| Scalar | Effect |
|---|---|
> 1 | Stretches the vector |
0 < s < 1 | Squishes the vector |
negative | Flips direction, then scales |
Numerically:
c ยท [x, y] = [cx, cy]
Multiply every component by the scalar.
Why "scalar"? Because it scales the vector โ Grant's way of making the name intuitive.
๐ก Key Takeaways
- Vectors live with their tail at the origin in linear algebra (unlike physics)
- Numbers โ arrows: you need both mental models
- Addition = chaining steps; scalar multiplication = stretching/flipping
- These two operations are the building blocks of everything in linear algebra
๐ Next in Series
โChapter 2: Linear combinations, span, and basis vectors