Linear Algebra
2.18 min read

Vectors as Data Packages

A vector is simply an ordered list of real numbers. It packages multiple quantities into a single object that we can manipulate mathematically. We write v=(v1,v2,…,vm)\mathbf{v} = (v_1, v_2, \ldots, v_m) where viv_i is the ii-th entry (or component).

The key word is ordered — (3,5)(3, 5) and (5,3)(5, 3) are different vectors. The order of entries matters because each position corresponds to a specific quantity (the first entry might be "x-coordinate," the second "y-coordinate").

Vectors in Rm\mathbb{R}^m are the fundamental objects of linear algebra. We'll see that operations on vectors — addition and scalar multiplication — have beautiful geometric interpretations that make abstract algebra visible.

Formal View

Definition 2.1 — Vector
A vector v∈Rm\mathbf{v} \in \mathbb{R}^m is an ordered mm-tuple of real numbers:
v=(v1v2⋮vm)\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_m \end{pmatrix}
The zero vector 0∈Rm\mathbf{0} \in \mathbb{R}^m has all entries equal to zero.

Why This Matters

Vectors encode any collection of related quantities as a single manipulable object.

  • A 3D position in space is a vector (x,y,z)∈R3(x, y, z) \in \mathbb{R}^3
  • An RGB color is a vector (r,g,b)∈R3(r, g, b) \in \mathbb{R}^3
  • One second of audio at 44,000 Hz is a vector in R44000\mathbb{R}^{44000}
  • A movie's ratings across 100 users is a vector in R100\mathbb{R}^{100}

Quiz

Question 1

The vectors (1,2,3)(1, 2, 3) and (3,2,1)(3, 2, 1) are equal.

Question 2

A vector in R5\mathbb{R}^5 has how many entries?

Common Mistakes

  • Thinking vectors must be "arrows" — they are just ordered lists of numbers.
  • Confusing R2\mathbb{R}^2 vectors (2 entries) with 2D points — they are mathematically the same thing.