Foundations — Orthogonal sets and orthonormal basis

Before you can enjoy the parent note, you need to own every mark it writes. We build each one from a picture, in an order where nothing appears before it is earned.

1. The arrow: what a vector $\mathbf{v}$ is

Plain words. $v_1$ is "how far right," $v_2$ is "how far up." Stack them and you have pinned down one arrow from the origin.

The picture. Start at the corner point (the origin) and walk $v_1$ steps east, then $v_2$ steps north. The arrow from start to finish is the vector.

Why the topic needs it. The whole chapter talks about sets of vectors $\{{\mathbf{u}}_1,\dots ,{\mathbf{u}}_p\}$. If "arrow = stack of numbers" is not automatic, none of the later formulas mean anything.

2. ${\mathbb{R}}^n$: the space the arrows live in

The picture. ${\mathbb{R}}^2$ is a flat sheet of paper. ${\mathbb{R}}^3$ is the room you sit in. ${\mathbb{R}}^n$ for bigger $n$ we can't draw, but every rule we prove with pictures in 2D still holds — that's the payoff of doing algebra.

Why the topic needs it. The parent writes "a set in ${\mathbb{R}}^n$." That is just "a bunch of arrows in flat $n$-dimensional space."

3. The dot product $\mathbf{u}\cdot \mathbf{v}$ — the star of the show

Why THIS tool and not another? We need one number that reports how much two arrows point the same way. Ordinary multiplication only works on numbers; the dot product is the machine built exactly to combine two arrows into an "agreement score." It answers the question "are these two arrows aligned, opposed, or perpendicular?"

Reading the sign — all cases, no gaps:

• $\mathbf{u}\cdot \mathbf{v}>0$ → angle less than $90^{\circ }$, arrows broadly agree.
• $\mathbf{u}\cdot \mathbf{v}=0$ → angle exactly $90^{\circ }$, arrows perpendicular. ⭐ This is the case the whole chapter is built on.
• $\mathbf{u}\cdot \mathbf{v}<0$ → angle more than $90^{\circ }$, arrows broadly oppose.
• If either arrow is the zero vector, the dot product is $0$ by the formula, but there is no angle — a degenerate case we keep in mind.

Why the topic needs it. "Orthogonal" is defined as ${\mathbf{u}}_i\cdot {\mathbf{u}}_j=0$. And the magic coordinate formula is nothing but a pile of dot products. See Dot product and norms for the full tour.

4. Length $\Vert \mathbf{v}\Vert$ — the norm

The picture. In the plane, $\mathbf{v}=\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]$ is the hypotenuse of a right triangle with legs $v_1$ and $v_2$. Pythagoras gives $\Vert \mathbf{v}\Vert =\sqrt{v_1^2+v_2^2}$.

Why the topic needs it. "Unit vector" and "normalize" are defined through length. And the coordinate denominator ${\mathbf{u}}_j\cdot {\mathbf{u}}_j$ is just $\Vert {\mathbf{u}}_j{\Vert }^2$.

5. Unit vector and "normalize"

Why divide by the length? Because $\Vert \frac{\mathbf{v}}{\Vert \mathbf{v}\Vert }\Vert =\frac{\Vert \mathbf{v}\Vert }{\Vert \mathbf{v}\Vert }=1$. The one forbidden input is $\mathbf{v}=0$: you cannot divide by zero length, and the zero arrow has no direction.

Why the topic needs it. "Orthonormal" = orthogonal plus every vector normalized. Example B in the parent literally builds ${\mathbf{q}}_1=\frac{1}{\sqrt{11}}{\mathbf{u}}_1$.

6. Perpendicular = orthogonal

Why a new word? "Perpendicular" feels tied to drawings in 2D. "Orthogonal" reminds us the test ($\mathbf{u}\cdot \mathbf{v}=0$) works in any dimension where we can't draw. Same concept, wider reach.

7. Linear combination, span, and basis

The picture. Each $c_i{\mathbf{u}}_i$ stretches arrow ${\mathbf{u}}_i$ by $c_i$; laying them tip-to-tail lands you at a new point.

Why the topic needs it. The whole chapter's headline — "an orthonormal basis is the dream coordinate system" — assumes you know a basis is just a coordinate system, and that Section 2 of the parent proves orthogonal sets are automatically independent.

8. Coordinates $c_j$

The picture. Coordinates are the "instructions" — how far along each axis to walk to reach $\mathbf{y}$. For a generic basis you find them by solving a system; for an orthogonal one you get them free by dotting. That contrast is the entire point of the topic. See Orthogonal projection for the geometric meaning of one such coordinate.

9. Matrix, transpose, and $U^{\mathsf{T}}U$

For an orthonormal set that grid is $1$ down the diagonal and $0$ everywhere else — the identity matrix $I$. That single fact powers QR decomposition and Least squares.

10. How it all feeds the topic

Everything on the left is a prerequisite; the arrows show what each idea unlocks. The parent topic sits at the bottom, fed by orthonormal bases and the identity $U^{\mathsf{T}}U=I$.

Equipment checklist

Cover the right side and test yourself.