Linear Algebra

Linear Transformations

A linear transformation is a function that preserves addition and scaling; in coordinates it becomes a matrix, determined by what it does to a basis.

status: reviewimportance: importantdifficulty 3/5math: undergraduateread: 16mdemo planned

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Concept Structure

Linear Transformations

01Intuition

Start with the picture, metaphor, or geometric mechanism.

02Math

Make the objects explicit and connect them with notation.

03Code

Mirror the equations with runnable implementation details.

04Interactive Demo

Manipulate the mechanism and watch the idea respond.

2prerequisites

2next concepts

2related links

Learning map

Linear Transformations

BeforeVector SpacesNow3/4 sections readyTryUse the demo notes to predict the mechanism before moving on.NextRepresentation Learning & Embedding Geometry

Object flow

3/4 sections readyAsk about this Research room

ConceptLinear TransformationsLinear Algebra EquationLinear Transformations equation 1Exact equation object CodeLinear Transformations code witness 1Exact code witness ClaimLinear Transformations central claimMechanism claim EquationLinear Transformations math objectsEquation and notation CodeLinear Transformations code witnessRunnable code object

ConceptLinear TransformationsLinear Algebra

Local snapshot ready

concept:linear-algebra/linear-transformations

Codewitness nearby Predictbefore reveal Roomobject handoff

Conceptual Bridge

What should feel connected as you move through this page.

Carry inVector Spaces

Bring the mental model from Vector Spaces; this page will reuse it instead of restarting from zero.

Work hereLinear Transformations

A linear transformation is a function that preserves addition and scaling; in coordinates it becomes a matrix, determined by what it does to a basis.

Carry outRepresentation Learning & Embedding Geometry

The next edge should feel earned: use the demo prediction here before following Representation Learning & Embedding Geometry.

Test the linkUse the demo notes to predict the mechanism before moving on.Then continue to Representation Learning & Embedding Geometry

01IntuitionStart with the picture, metaphor, or geometric mechanism.02MathMake the objects explicit and connect them with notation.03CodeMirror the equations with runnable implementation details.04Interactive DemoManipulate the mechanism and watch the idea respond.

Intuition

Build the mental picture first so the rest of the page has something to attach to.

Section prompt

A linear transformation is a “machine for vectors” that respects the two operations that define a vector space:

if you add two inputs, the outputs add the same way
if you scale an input, the output scales the same way

That sounds restrictive, but it’s exactly what makes linear maps easy: once you know what the machine does to a basis, you know what it does to every vector.

This is the bridge from geometry to matrices: a matrix is just a linear transformation written in coordinates.

In deep learning, learned matrices are everywhere: embedding projections, MLP layers, attention query/key/value maps, and feature-space rotations all use the same idea. The nonlinear parts of a network are important, but the linear maps are the pieces that move information between coordinate systems.

Math

Translate the story into symbols, assumptions, and a derivation you can inspect.

Section prompt

Equation 1T(u+v) = T(u) + T(v), \qquad T(\alpha v) = \alpha T(v).Equation 2T(x) = Ax

Definition. A function $T: V \to W$ is linear if for all $u,v \in V$ and scalars $\alpha$ :

T(u+v) = T(u) + T(v), \qquad T(\alpha v) = \alpha T(v).

In $\mathbb{R}^n$ with the standard basis, every linear map has the form

T(x) = Ax

for some matrix $A$ .

A key concrete fact: the columns of $A$ are where the basis vectors go. If $e_i$ is the $i$ th standard basis vector, then

A e_i = \text{column}_i(A).

Two structural ideas show up everywhere:

Composition corresponds to matrix multiplication: $(S\circ T)(x) = S(T(x))$ becomes $(BA)x$ .
Kernel and image: $\ker(T)=\{x:Ax=0\}$ and $\operatorname{im}(T)=\{Ax:x\in V\}$ .

In attention layers, for example, token representations $x$ are multiplied by learned matrices:

q = W_Qx,\qquad k = W_Kx,\qquad v = W_Vx.

Those are linear transformations into query, key, and value spaces. The dot products happen after the representations have been moved into those learned coordinate systems.

Code

Keep the implementation aligned with the notation so the algorithm is legible.

Section prompt

Code witness 1import numpy as np A = np.array([ [1.2, 0.3], [-0.4, 0.8], ]) e1 = np.array([1.0, 0.0]) e2 =...python

import numpy as np

A = np.array([
    [1.2, 0.3],
    [-0.4, 0.8],
])

e1 = np.array([1.0, 0.0])
e2 = np.array([0.0, 1.0])

print("A e1 =", A @ e1)  # column 1
print("A e2 =", A @ e2)  # column 2

x = np.array([2.0, -1.0])
print("T(x) =", A @ x)
print("area scale det(A) =", np.linalg.det(A))

Interactive Demo

Use direct manipulation to connect the explanation to a moving system.

Section prompt

No interactive demo yet. A great demo: drag a matrix (shear/rotate/scale) and watch a grid + basis vectors transform, plus show determinant as area scaling.

No live visualization is registered for this concept yet.

The page still supports explanatory demo notes above; when a viz.tsx exists, it will render here without changing the route.

After The First Pass

Turn the concept into an inspected object.

Once the invariant is visible in the intuition, math, code, and demo, use these panels to inspect the mechanism visually, check source support, practice the idea, and attach a grounded research question.

Mechanism Storyboard

See the idea move before the page explains it

A linear transformation is a function that preserves addition and scaling; in coordinates it becomes a matrix, determined by what it does to a basis.

Demo notes open01 / Intuition

Prediction lens

Start with the picture, metaphor, or geometric mechanism.

Commit first

Before reading further, choose the kind of change Linear Transformations should make visible.

Visual Inquiry

Make the image answer a mathematical question

A linear transformation is a function that preserves addition and scaling; in coordinates it becomes a matrix, determined by what it does to a basis.

3/4 stages readyDemo notes connected

Visual cueWhich visible object should carry the first intuition?

Inspection depth2/4

Prediction

Which visible object should carry the first intuition?

Commit first

Pick the cue that should make Linear Transformations easier to reason about before the page gives the answer.

Claim Review

A linear transformation is a function that preserves addition and scaling; in coordinates it becomes a matrix, determined by what it does to a basis.

StatusSubstantive claim review pending

Source IDs and witness objects are attached for review; they are not proof by themselves.

SourcesNo references

Add source metadata before claiming support.

Witnesses3 local objects

Use equation, code, and demo objects to check whether the source support is operational.

Source support candidates

No structured source note is attached yet.

Mechanism witnesses

Equation 1

T(u+v) = T(u) + T(v), \qquad T(\alpha v) = \alpha T(v).

Equation 2

T(x) = Ax

Code witness 1import numpy as np A = np.array([ [1.2, 0.3], [-0.4, 0.8], ]) e1 = np.array([1.0, 0.0]) e2 =...

Practice Loop

Try the idea before it explains itself

A linear transformation is a function that preserves addition and scaling; in coordinates it becomes a matrix, determined by what it does to a basis.

Readiness0/3 checks ready

Predict

Before touching the demo, predict one visible change that should happen in Linear Transformations.

Hint 1

Reveal when your model needs a nudge.

Hint 2

Reveal when your model needs a nudge.

Hint 3

Reveal when your model needs a nudge.

Your answer canvas

Local checks

Claim

A concrete answer is on the canvas.

Mechanism

The answer names why the claim should hold.

Bridge

It touches the page context or a neighboring idea.

Misconception check

Object research drawerClose

ConceptLinear TransformationsLinear Algebra

Code witness comparisonLinear Transformations code witness 1A = np.array([Prediction before revealLinear Transformations predictionUse the demo notes to predict the mechanism before moving on.

Grounded room questionWhat is the smallest example that makes Linear Transformations click without losing the math?Local snapshot ready

Research Room

Attach the question to an exact object

Pick the concept, equation, source, code witness, claim, misconception, or demo state before asking for help. The handoff stays grounded to that object.

Next local actionNo local draft saved yet

Open the draft below to save one note and next action in this browser.

conceptLinear Algebra

Linear Transformations

Anchored question

What is the smallest example that makes Linear Transformations click without losing the math?

Local action draftNo local draft saved yetExpand only when ready to capture one local next action

Local action draft

This draft stays locally in this browser for concept:linear-algebra/linear-transformations.

Draft noteNext action

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Evidence to inspect

Definition, prerequisite, and contrast concept links
The equation or code witness that makes the concept operational
One demo state that shows the invariant instead of a slogan

What would resolve this

The learner can state the mechanism in their own words
The learner can name the prerequisite that would repair confusion
The learner can predict how the mechanism changes under one perturbation

Grounded AI handoff

I am working in Continuous Function's research reading room. Object: concept - Linear Transformations Object key: concept:linear-algebra/linear-transformations Context: Linear Algebra Anchor id: concept/concept-notebook/linear-algebra/linear-transformations Open question: What is the smallest example that makes Linear Transformations click without losing the math? Evidence to inspect: - Definition, prerequisite, and contrast concept links - The equation or code witness that makes the concept operational - One demo state that shows the invariant instead of a slogan What would resolve this: - The learner can state the mechanism in their own words - The learner can name the prerequisite that would repair confusion - The learner can predict how the mechanism changes under one perturbation Answer as a careful research tutor: stay source-grounded, separate verified evidence from assumptions, name the relevant math objects, and end with one next action.

Open source object

concept/concept-notebook/linear-algebra/linear-transformations
concept:linear-algebra/linear-transformations

Learning Map

Before / Now / Try / Next

BeforeVector Spaces

NowIntuition → Math → Code → Demo

TryUse the demo notes to predict the mechanism before moving on.

NextRepresentation Learning & Embedding Geometry

Intuitionready
Mathready
Codeready
Interactive Demoplanned

Object Companion

Ask beside the selected object

A linear transformation is a function that preserves addition and scaling; in coordinates it becomes a matrix, determined by what it does to a basis.

Your question

GoalComfortStyleStuck on

Context prompt

You are my AI learning companion for Continuous Function. Current context: Linear Algebra concept. Learning surface: Linear Transformations. What this page says: A linear transformation is a function that preserves addition and scaling; in coordinates it becomes a matrix, determined by what it does to a basis. Current section: Intuition, math, code, and interactive demo. Suggested next step: Use the demo notes to predict the mechanism before moving on.. Learner goal: Understand the idea. Learner comfort level: New to this. Preferred explanation style: Visual first. Task: Explain the central idea in plain language, then restate it with the exact math objects from the page. Answer in a way that helps me learn: ask one clarifying question only if needed, use intuition before notation, and end with one thing I should try on the page.

Domain

Linear Algebra

linear-algebramatricesfunctions

Prerequisites

Vector Spaces Basis and Span

Leads To

Representation Learning & Embedding Geometry Scaled Dot-Product Attention & Transformer Layers

Dot Product Norms

Within this domain