geometric-algebraic-bridge

What this skill does

# Geometric-Algebraic Bridge

## Table of Contents
1. [Workflow](#workflow)
2. [The Bridge Sentence](#the-bridge-sentence)
3. [Layout Templates](#layout-templates)
4. [Common Patterns](#common-patterns)
5. [Guardrails](#guardrails)
6. [Quick Reference](#quick-reference)

A geometric picture without the algebra leaves the learner unable to compute. An algebraic formula without the picture leaves them unable to *see*. This skill produces both, side by side, with one sentence between them that says — in plain English — why they are the same thing.

The bridge sentence is the load-bearing piece. Without it, you've handed the learner two unrelated halves and made them do the joining themselves.

**Quick example (Dot product):**

> **Geometric:** Two arrows in the plane. The dot product a · b measures *how much one points along the other* — the projection of a onto b's direction, scaled by b's length.
>
> **Algebraic:** a · b = a₁b₁ + a₂b₂ + … + aₙbₙ. (Or equivalently, a · b = |a| |b| cos θ.)
>
> **Bridge:** The component-wise sum *is* the projection-times-length. You can prove it by writing a in coordinates aligned with b — the sum collapses to |a|cos θ × |b|, which is what projection-times-length says geometrically.

Three short blocks. The bridge is the last, and the only one that's not optional.

## Workflow

Copy this checklist and track your progress:

```
Bridge Progress:
- [ ] Step 1: Identify the concept and which view the learner has
- [ ] Step 2: Construct the missing view, kept short and concrete
- [ ] Step 3: Write the bridge sentence — one sentence, plain English
- [ ] Step 4: Verify the bridge with a small example
- [ ] Step 5: Invite the learner to confirm or push back
```

**Step 1: Identify the concept and which view the learner has**

Most users come with one view, not zero. Diagnose which:

- **Algebra-first learner** (most ML practitioners): "I understand the equation Av = λv but I can't see what it *means*." → Build the geometric view.
- **Geometry-first learner** (visual / physics-trained): "I get the picture of a rotation, but I don't know how to write it down." → Build the algebraic view.
- **Neither** (rarer): use the `concept-rediscovery-walk` skill instead — they need to invent the concept first.

A diagnostic question that almost always tells you which: "Can you predict what this concept does without computing? Or do you need to compute first?" Predict-without-compute = geometric; compute-first = algebraic.

**Step 2: Construct the missing view, kept short and concrete**

Build the missing view in 3-5 sentences. Length is a discipline: longer = more chance the learner loses the thread before the bridge.

- **For the geometric view:** Use a specific example (a 2×2 matrix, a small transformation, a labeled diagram). Prefer 2D unless the concept genuinely requires 3D.
- **For the algebraic view:** Show the formula, then *say what it does, term by term, in 1-2 sentences*. Don't drop the formula and walk away.

For per-concept layout templates, see [resources/templates.md](resources/templates.md). For full worked examples, see [resources/examples.md](resources/examples.md).

**Step 3: Write the bridge sentence — one sentence, plain English**

The bridge is the highest-leverage sentence in the response. It must:
- Be one sentence (two if absolutely necessary).
- Use plain English — no LaTeX, no symbols if avoidable.
- Say *why* the two views are the same thing — not just *that* they are.

Three bridge formulas that work for most concepts:

- **"X *is* Y, because…"** — direct identification. ("The dot product *is* the projection times length, because writing a in coordinates aligned with b collapses the sum to |a|cos θ × |b|.")
- **"X happens *because* Y."** — causal. ("Eigenvectors get only stretched, not rotated, because they're the directions where Av = λv — and λv is just a rescaling.")
- **"X is what Y looks like when you do Z."** — perspective shift. ("A matrix is what a linear transformation looks like once you pick a basis.")

The bridge fails when it just restates one view in symbols and the other in words without explaining the *why*. "Av = λv means v is left in place up to scaling" is not a bridge; it's a translation. The bridge is "and *that's why* it's the special direction the transformation can't rotate".

**Step 4: Verify the bridge with a small example**

After the bridge, run *one* small example end-to-end through both views:

- "Take A = [[2, 0], [0, 3]]. Geometrically, it stretches the x-axis by 2, the y-axis by 3. Algebraically, eigenvalues are 2 and 3, eigenvectors are (1, 0) and (0, 1). Same picture, same numbers — bridge confirmed."

The verification example should fit on one line if possible. Its purpose is to make the bridge *concrete* — not to teach a new concept.

**Step 5: Invite the learner to confirm or push back**

End with one of:
- "Does the bridge land? Or is there a step that still feels arbitrary?"
- "Try [small variant]. Predict the result both ways before computing."
- "What's still unclear?"

This catches the case where the bridge made sense to *you* but not to the learner. Bridges are subjective; the learner is the only judge.

## The Bridge Sentence

The bridge is a single sentence (or two) that explicitly identifies the geometric and algebraic descriptions as the same thing, with a *because*. It is the part of the explanation most often skipped, and the part most often missed by learners who say "I sort of get it but it doesn't click."

### What a good bridge does
- Names both views by their key noun. ("The eigenvector equation Av = λv corresponds to the *invariant direction* in the picture, *because* λv just rescales v without rotating it.")
- Includes the word "because" or its functional equivalent. The bridge is *causal*, not just declarative.
- Uses the most concrete language available. Prefer "stretched" over "scaled by a positive scalar"; prefer "the picture of an ellipsoid" over "a positive semidefinite quadratic form".

### What a bad bridge does
- Restates the formula. ("Av = λv means v is an eigenvector of A.") That's a definition, not a bridge.
- Hedges. ("These are essentially related…", "in some sense the same thing…") If you can't say *why*, you don't have a bridge yet.
- Skips the geometric noun. ("Eigenvectors satisfy Av = λv.") Where's the picture?

For a library of bridge sentences per concept, see [resources/templates.md](resources/templates.md).

## Layout Templates

### Template A: Bridge after both views (default)

```
**Geometric view.** [3-5 sentence picture, with one specific example.]

**Algebraic view.** [Formula, with 1-2 sentences saying what each term does.]

**Bridge.** [One sentence: X *is* Y because Z.]

**Verify on a tiny example.** [One-line check that both views give the same answer.]
```

Use when both views are roughly equal weight. This is the default.

### Template B: Lead with the missing view

```
**[The view the learner has].** You already have this — [restate in one sentence].

**[The missing view].** [3-5 sentences, ending in the formula or the picture.]

**Bridge.** [One sentence.]

**Verify.** [One line.]
```

Use when the learner has one view confidently and you don't want to belabor it.

### Template C: Three-column for compound concepts

```
| Geometric | Algebraic | Bridge |
|---|---|---|
| [Sentence] | [Formula] | [Why same] |
```

Use only for compound concepts with multiple sub-bridges (e.g., SVD = rotate, scale, rotate; attention = ask, match, mix). The table prevents the bridge from getting lost in prose.

For full filled-in templates per concept, see [resources/templates.md](resources/templates.md).

## Common Patterns

### Concepts where the bridge is the whole game
- Eigenvectors (formula ↔ invariant directions)
- Dot product (sum ↔ projection × length)
- Determinant (formula ↔ signed area scaling)
- Covariance matrix (formula ↔ ellipsoidal shape of data cloud)
- Softmax (formula ↔ point on the simplex; sharp at high contrast)
- Cr