Foundations — Basis — definition, uniqueness of representation

Before you can trust that sentence, you need to own every symbol the parent note throws at you. This page builds each one from nothing, in an order where every piece rests on the piece before it. If a smart 12-year-old reads from line one, they should never hit a symbol they haven't already met.

1. What is a vector, really? (the symbol $v$)

A vector is anything you can do two things with: add two of them, and scale one bigger or smaller. That's the whole job description.

The picture that makes this concrete: an arrow starting at the origin (the point $(0,0)$) and pointing to some spot.

An entry like $v_1$ is called a component — one number telling you how far to go along one axis. Notice: a component is tied to which axes you drew. Change the axes, the components change even though the arrow stayed put. Hold that thought — it returns in Section 6.

2. The number that scales: a scalar (the symbol $c$)

When we write $cv$, the letter $c$ is a plain number — we call it a scalar because its job is to scale (stretch, shrink, or flip) the vector.

Cover every case so you're never surprised:

• $c>1$: longer, same direction.
• $0<c<1$: shorter, same direction.
• $c=0$: shrinks to the origin (the zero vector, next section).
• $c<0$: flips to the opposite side, length $|c|$ times original.

3. The special arrow that goes nowhere: $0$

4. Combining arrows: the sum $\sum$ and a linear combination

The parent writes ${\sum }_i{c}_i{v}_i$. Let's unpack that symbol completely.

Reading it aloud: "scale arrow $v_1$ by $c_1$, scale $v_2$ by $c_2$, …, then add all the scaled arrows tip-to-tail."

The counter starts at $i=1$ and stops at $i=n$; that top number $n$ is how many arrows are in the set — which becomes the dimension. See Span and Spanning Sets for the "reach everything" side and Linear Independence for the "no redundancy" side.

5. The container: a vector space $V$ and a set $B$

The symbol ${\mathbb{R}}^2$ reads "R-two": lists of 2 real numbers. ${\mathbb{R}}^3$: lists of 3. The superscript is how many components each vector carries. See Subspaces for smaller spaces living inside a bigger one.

6. Coordinates: the symbol $[w{]}_B$

Once you fix a basis $B$, the scalars in $w=c_1{v}_1+\cdots +c_n{v}_n$ get a name and a notation.

Case check: with the standard basis $\{e_1,e_2\}$, the coordinates equal the components — that's why the standard basis "feels invisible." With any tilted basis they differ, which is exactly the parent's $(4,2)\to (3,1)$ example.

7. Reading the logic symbols: $⟹$, $⟺$, $\forall$

The proofs lean on three logic shorthands. Own them so the proof reads like English.

8. The two verbs: span and independent (in plain words)

You now have every symbol needed to state the two basis conditions in words.

The size of a basis — the top number $n$ in every $\sum$ — is the dimension, explored in Dimension. And feeding basis arrows through a transformation builds Matrix of a Linear Map.

Prerequisite map

Equipment checklist

Test yourself — reveal only after you've answered aloud.

Connections

• Linear Independence — the "no redundancy" verb built here in words.
• Span and Spanning Sets — the "reach everything" verb built here in words.
• Dimension — the number $n$ of arrows in a basis.
• Coordinate Vectors and Change of Basis — where $[w{]}_B$ takes centre stage.
• Subspaces — smaller vector spaces living inside a bigger $V$.
• Matrix of a Linear Map — built by feeding basis vectors through a map.
• Basis — definition, uniqueness of representation — the parent this page equips you for.