Linear Algebra

Basis and Span

The span is everything you can build by mixing vectors; a basis is a non-redundant set that spans the whole space and gives unique coordinates.

status: reviewimportance: criticaldifficulty 2/5math: undergraduateread: 14mdemo planned

Back to Linear Algebra Next: Linear Transformations

Concept Structure

Basis and Span

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

1next concepts

2related links

Learning map

Basis and Span

BeforeVector SpacesNow3/4 sections readyTryUse the demo notes to predict the mechanism before moving on.NextLinear Transformations

Object flow

3/4 sections readyAsk about this Research room

ConceptBasis and SpanLinear Algebra EquationBasis and Span equation 1Exact equation object CodeBasis and Span code witness 1Exact code witness ClaimBasis and Span central claimMechanism claim EquationBasis and Span math objectsEquation and notation CodeBasis and Span code witnessRunnable code object

ConceptBasis and SpanLinear Algebra

Local snapshot ready

concept:linear-algebra/basis-and-span

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 hereBasis and Span

The span is everything you can build by mixing vectors; a basis is a non-redundant set that spans the whole space and gives unique coordinates.

Carry outLinear Transformations

The next edge should feel earned: use the demo prediction here before following Linear Transformations.

Test the linkUse the demo notes to predict the mechanism before moving on.Then continue to Linear Transformations

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

Think of vectors as “building blocks.” The span is the set of everything you can build by mixing those blocks with real-valued weights.

A basis is what you get when you remove all redundancy but still keep enough blocks to build everything in the space. When you have a basis, every vector has a unique coordinate description.

That uniqueness is why bases show up everywhere: once you pick a basis, a complicated abstract space turns into a list of numbers (coordinates), and linear algebra becomes computation.

Math

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

Section prompt

Equation 1\operatorname{span}(v_1,\dots,v_k) = \left\{\sum_{i=1}^k a_i v_i \;\middle|\; a_i \in \mathbb...Equation 2x = \sum_{i=1}^n c_i b_i.

Span. Given vectors $v_1,\dots,v_k$ in a vector space $V$ , their span is

\operatorname{span}(v_1,\dots,v_k) = \left\{\sum_{i=1}^k a_i v_i \;\middle|\; a_i \in \mathbb{R} \right\}.

Basis. A set $\{b_1,\dots,b_n\}$ is a basis of $V$ if:

it spans $V$ , and
it is linearly independent.

Key insight: in a basis, coordinates are unique. If $x \in V$ and $\{b_i\}$ is a basis, then there exists a unique coefficient vector $c \in \mathbb{R}^n$ such that

x = \sum_{i=1}^n c_i b_i.

Matrix form: let $B = [b_1\;\cdots\; b_n]$ . Then

x = Bc.

If $B$ is square and invertible (a basis in $\mathbb{R}^n$ ), then

c = B^{-1}x.

Code

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

Section prompt

Code witness 1import numpy as np b1 = np.array([1.0, 1.0]) b2 = np.array([1.0, -1.0]) B = np.column_stack([...python

import numpy as np

b1 = np.array([1.0, 1.0])
b2 = np.array([1.0, -1.0])
B = np.column_stack([b1, b2])  # basis matrix

x = np.array([3.0, 1.0])

c = np.linalg.solve(B, x)
print("coordinates c:", c)
print("reconstruct:", B @ c)

Interactive Demo

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

Section prompt

No interactive demo yet. A good demo: choose 2 vectors in 2D, watch the span (a line vs the whole plane) and whether the basis matrix becomes singular.

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

The span is everything you can build by mixing vectors; a basis is a non-redundant set that spans the whole space and gives unique coordinates.

Demo notes open01 / Intuition

Prediction lens

Start with the picture, metaphor, or geometric mechanism.

Commit first

Before reading further, choose the kind of change Basis and Span should make visible.

Visual Inquiry

Make the image answer a mathematical question

The span is everything you can build by mixing vectors; a basis is a non-redundant set that spans the whole space and gives unique coordinates.

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 Basis and Span easier to reason about before the page gives the answer.

Claim Review

The span is everything you can build by mixing vectors; a basis is a non-redundant set that spans the whole space and gives unique coordinates.

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

\operatorname{span}(v_1,\dots,v_k) = \left\{\sum_{i=1}^k a_i v_i \;\middle|\; a_i \in \mathbb{R} \right\}.

Equation 2

x = \sum_{i=1}^n c_i b_i.

Code witness 1import numpy as np b1 = np.array([1.0, 1.0]) b2 = np.array([1.0, -1.0]) B = np.column_stack([...

Practice Loop

Try the idea before it explains itself

The span is everything you can build by mixing vectors; a basis is a non-redundant set that spans the whole space and gives unique coordinates.

Readiness0/3 checks ready

Predict

Before touching the demo, predict one visible change that should happen in Basis and Span.

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

ConceptBasis and SpanLinear Algebra

Code witness comparisonBasis and Span code witness 1b1 = np.array([1.0, 1.0])Prediction before revealBasis and Span predictionUse the demo notes to predict the mechanism before moving on.

Grounded room questionWhat is the smallest example that makes Basis and Span 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

Basis and Span

Anchored question

What is the smallest example that makes Basis and Span 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/basis-and-span.

Draft noteNext action

No local draft saved.

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 - Basis and Span Object key: concept:linear-algebra/basis-and-span Context: Linear Algebra Anchor id: concept/concept-notebook/linear-algebra/basis-and-span Open question: What is the smallest example that makes Basis and Span 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/basis-and-span
concept:linear-algebra/basis-and-span

Learning Map

Before / Now / Try / Next

BeforeVector Spaces

NowIntuition → Math → Code → Demo

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

NextLinear Transformations

Intuitionready
Mathready
Codeready
Interactive Demoplanned

Object Companion

Ask beside the selected object

The span is everything you can build by mixing vectors; a basis is a non-redundant set that spans the whole space and gives unique coordinates.

Your question

GoalComfortStyleStuck on

Context prompt

You are my AI learning companion for Continuous Function. Current context: Linear Algebra concept. Learning surface: Basis and Span. What this page says: The span is everything you can build by mixing vectors; a basis is a non-redundant set that spans the whole space and gives unique coordinates. 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-algebrabasiscoordinates

Prerequisites

Vector Spaces Linear Independence

Leads To

Linear Transformations

Dot Product Norms

Within this domain