Foundations — Abstract vector spaces — axioms, examples beyond ℝⁿ

Before you can read the parent note, you must be able to read its alphabet. This page walks every symbol and idea it uses, in the order they build on each other. Nothing is assumed — we start below zero.

The symbols, built one on top of the next

1. A set and the symbol $\in$

Picture: a box. If $v$ is a thing in the box $V$, we write $v\in V$ ("$v$ is in $V$"). If it's outside, $v\notin V$.

Why the topic needs it: the whole definition of a vector space begins "a set $V$ …". Every later idea (closure, zero, basis) is a statement about what is or isn't inside the box.

2. Sets of numbers: $\mathbb{R}$, $\mathbb{C}$, and a field $\mathbb{F}$

Picture: $\mathbb{R}$ is a straight ruler; $\mathbb{C}$ is a flat sheet of graph paper with a real axis and an imaginary axis.

Why this tool and not just "numbers"? The parent note scales vectors — it multiplies them by numbers. We need to name which numbers are legal to multiply by. Those numbers form the field $\mathbb{F}$. The choice matters: the same set $\mathbb{C}$ has dimension 2 over $\mathbb{R}$ but dimension 1 over $\mathbb{C}$ — the field you scale with changes the answer. See Fields and scalars.

3. ${\mathbb{R}}^n$ — the box you already know

Picture: an arrow from the origin to the point $(3,1)$.

Why the topic needs it: ${\mathbb{R}}^n$ is the familiar box. The parent note's entire mission is to say "keep the rules of ${\mathbb{R}}^n$, throw away the arrows." So we must first know what the arrow-box does.

4. The two operations: $+$ and $\cdot$

Why these two and nothing else? Linear algebra is exactly the study of what you can build with only "add" and "scale". Multiplying two vectors together is not part of the definition — that extra structure (a dot product) lives in Inner product spaces, a separate upgrade.

5. Closure — the hidden axiom #0

Picture: the box has walls. Closure says every "add" and "scale" arrow lands back inside the walls.

Why the topic needs it: the parent calls this "axiom #0 and the one beginners forget." A shape can look like a vector space and fail here — e.g. a line not through the origin: scale a point on it by $0$ and you fall off the line.

6. The zero vector $0$ and the inverse $-v$

Why the topic needs it: axioms 3 and 4 are built entirely on these. The proofs in the parent ("$0v=0$", "$(-1)v=-v$") are just careful games with $0$ and $-v$.

7. Quantifier symbols: $\forall$, $\exists$, and $:$

Example read-aloud: $\exists 0$ with $v+0=v$ says "there exists a zero vector such that adding it changes nothing."

Why the topic needs it: every one of the 8 axioms is a $\forall$ or $\exists$ sentence. If you can't read these, the axiom list is gibberish.

8. The map notation $f:A\to B$

Picture: the parent writes $+:V\times V\to V$. Read it: "addition eats a pair of vectors ($V\times V$, two boxes side by side) and returns one vector in $V$." That final "$\to V$" is closure written in symbols!

Why the topic needs it: this notation is how closure is stated formally, and it's the language of Linear maps and matrix representations later.

9. Dimension and basis (preview)

Picture: ${\mathbb{R}}^2$ needs 2 arrows (right, up) to reach anywhere → dimension 2. A box of degree-$\le 2$ polynomials needs $\{1,x,x^2\}$ → dimension 3.

Why the topic needs it: the parent lists dimensions ($P_n$ has $n+1$, $M_{m\times n}$ has $mn$, $C[0,1]$ has $\infty$). You don't need to master this yet — that's Linear independence, basis and dimension — but you must know the words mean "count of building blocks."

10. The example boxes' notation

Why the topic needs it: these are the payoff — proof that "vector" need not mean "arrow". You only need to recognise the notation; the parent supplies the rest.

How these foundations feed the topic

Equipment checklist

Read each question, answer in your head, then reveal.

Connections

• Parent: Abstract vector spaces
• Fields and scalars
• Subspaces and the subspace test
• Linear independence, basis and dimension
• Linear maps and matrix representations
• Inner product spaces
• Function spaces and Fourier series