algebra

Vector Spaces

Here is the truth about vector spaces that no textbook says plainly enough: a vector is anything you can add to another of its kind and stretch by a number. That’s it. Arrows are just the most famous example.

Once you see that, functions, polynomials, sound waves and quantum states all turn out to be vectors.

The core idea, in plain words

Pick a Field $F$ of scalars (usually $\mathbb{R}$ or $\mathbb{C}$ ). A vector space over $F$ is a set $V$ with:

addition $u + v$ , making $(V, +)$ an abelian Group with a zero vector $\mathbf{0}$ , and
scalar multiplication $c\,v$ for $c \in F$ ,

satisfying the natural rules: $c(u + v) = cu + cv$ , $(c + d)v = cv + dv$ , $(cd)v = c(dv)$ and $1v = v$ .

Exactly the definition of a Module, with the ring promoted to a field. And that promotion buys a miracle: every vector space has a basis, a set of vectors such that every vector is a combination of them in exactly one way. The number of basis vectors is the same however you choose them, and it is called the dimension.

An everyday analogy: colours on a screen

Every colour your screen shows is a mix of red, green and blue light. “Add” two colours by adding their light; “scale” a colour by dimming or brightening it. Red, green and blue form a basis: every colour is a unique recipe of so much red, so much green, so much blue. Colour space is 3-dimensional. You could choose a different basis (cyan, magenta, yellow, say) and describe the same colours with different numbers, but you’d always need exactly three.

A worked example: polynomials as vectors

Let $P_2$ be all polynomials of degree at most 2 with real coefficients, $a + bx + cx^2$ . Add two of them and you get another; multiply by a real number and you get another. So $P_2$ is a vector space.

A basis: $\{1, x, x^2\}$ . The polynomial $3 - 2x + 5x^2$ has coordinates $(3, -2, 5)$ . So $\dim P_2 = 3$ , and $P_2$ behaves exactly like $\mathbb{R}^3$ .

Now check a different basis: $\{1,\ 1 + x,\ 1 + x + x^2\}$ . Write $3 - 2x + 5x^2$ in it:

3 - 2x + 5x^2 = a \cdot 1 + b(1 + x) + c(1 + x + x^2).

Matching $x^2$ : $c = 5$ . Matching $x$ : $b + c = -2$ , so $b = -7$ . Matching constants: $a + b + c = 3$ , so $a = 5$ . New coordinates $(5, -7, 5)$ . Same vector, different coordinates. The vector doesn’t care which basis you use; only its description changes.

And linear independence: are $1 + x$ , $1 - x$ and $2$ independent? No: $(1 + x) + (1 - x) - 2 = 0$ . One of them is redundant, and together they only span a 2-dimensional piece of $P_2$ .

A picture in words: a flat sheet through the origin

Stand in ordinary 3D space. A subspace is a flat thing through the origin: the origin alone (dimension 0), a line through it (dimension 1), a plane through it (dimension 2), or everything (dimension 3). A plane that misses the origin is not a subspace, because it doesn’t contain $\mathbf{0}$ , and adding two of its vectors pushes you off it.

Two arrows not on the same line span a plane: every point on the plane is “so much of the first plus so much of the second”, like coordinates on a slanted sheet of graph paper.

graph LR
  S["Spanning set: reaches everything"] --> B["Basis: spanning AND independent"]
  I["Independent set: no redundancy"] --> B
  B --> D["Dimension = size of any basis"]

Common misconceptions

“Vectors are lists of numbers.” Lists are what you get after choosing a basis. The vector itself exists before you pick coordinates.
“Vectors have length and direction.” Only once you add an inner product (Inner Product Spaces). A plain vector space has no notion of length or angle.
“Any plane is a vector space.” Only planes through the origin.
“Infinite-dimensional is just big.” It’s different. The space of all continuous functions has no finite basis, and questions like “does this infinite sum converge?” enter. That’s where analysis and quantum mechanics live.

How it connects

Down to Fields (the scalars), Groups (vector addition) and Modules (the ring version).
Sideways to Linear Algebra, the overview of the whole subject.
Up to Linear Operators and Matrices, the machines between vector spaces, and to Inner Product Spaces, Tensors and Exterior Algebras.
Out to Quantum Mechanics, where states are vectors in a complex vector space and superposition is just vector addition.

The truth revealed: a vector is not an arrow. It is anything that adds and scales, and arrows are only the first thing that does.