Prerequisite · Self-Directed · Beginner to Intermediate

CALCULUS &
LINEAR
ALGEBRA

A structured self-study curriculum covering single-variable calculus through multivariable methods, and linear algebra from systems of equations to eigendecomposition. Modeled after MIT 18.01, 18.02, and 18.06 — the mathematical backbone for any engineering discipline.

18 Weeks

6 Modules

MIT OCW Referenced

3Blue1Brown Companion

Self-Paced

← Return to Robotics Path

MIT 18.01Single Variable Calculus

MIT 18.02Multivariable Calculus

MIT 18.06Linear Algebra

3Blue1BrownEssence of Linear Algebra

3Blue1BrownEssence of Calculus

Core Modules

Weeks

Textbooks

∞

Free Resources

Prerequisites

Algebra & Functions

Polynomials, factoring, rational expressions, exponents, logarithms. You must be comfortable manipulating equations before calculus will make sense.

Trigonometry

Unit circle, sine, cosine, tangent, identities. Trig functions appear everywhere in calculus and rotation matrices in linear algebra.

Arithmetic & Number Sense

Fractions, order of operations, basic proof reading. Comfort with mathematical notation and logical reasoning will carry you through every module.

Limits & Differentiation

MODULE 01A

WEEKS 1–2

Limits & Continuity

The concept that underpins all of calculus. What happens as a value approaches something — and why "approaching" and "arriving" are different ideas. MIT 18.01 starts here.

Intuitive definition of a limit

One-sided limits and squeeze theorem

Epsilon-delta definition (formal)

Continuity and types of discontinuity

Limits at infinity and asymptotic behavior

MODULE 01B

WEEKS 2–3

Derivatives & Differentiation Rules

The derivative measures instantaneous rate of change. From the limit definition to mechanical rules, this is the core tool of single-variable calculus.

Derivative as a limit (first principles)

Power, product, quotient rules

Chain rule and implicit differentiation

Derivatives of trig, exponential, log functions

Higher-order derivatives

TEXTBOOKS FOR MODULE 01

Calculus: Early Transcendentals

James Stewart · Cengage, 8th Ed.

The standard university calculus textbook. Clear explanations, excellent exercises.

Core

Calculus

Michael Spivak · Publish or Perish, 4th Ed.

Rigorous approach for those who want to understand the "why" behind every theorem.

Rigorous

Applications & Integration

MODULE 02A

WEEKS 4–5

Applications of Derivatives

Using derivatives to understand curves, find extrema, and solve optimization problems. This is where calculus starts to feel powerful.

Mean Value Theorem

Curve sketching: increasing/decreasing, concavity

Local and global extrema

Optimization problems

Related rates

L'Hôpital's Rule

MODULE 02B

WEEKS 5–6

Integration

The reverse of differentiation — and far more than that. The Fundamental Theorem of Calculus connects area under curves to antiderivatives.

Antiderivatives and indefinite integrals

Riemann sums and definite integrals

Fundamental Theorem of Calculus (Parts I & II)

Substitution (u-sub)

Integration by parts

Applications: area, volume, arc length

TEXTBOOKS FOR MODULE 02

Calculus: Early Transcendentals

James Stewart · Chapters 4–7

Continues from Module 01. Applications and integration techniques.

Core

Thomas' Calculus

Hass, Heil, Weir · Pearson, 14th Ed.

Alternative to Stewart with strong problem sets. Used in many US universities.

Alternative

Systems of Equations & Matrix Algebra

MODULE 03

WEEKS 7–9

Matrices, Vectors & Linear Systems

The starting point of linear algebra. How to represent and solve systems of equations with matrices, and why this matters for everything from computer graphics to machine learning. MIT 18.06 builds from here.

Vectors in R² and R³, dot and cross product

Systems of linear equations

Gaussian elimination and row echelon form

Matrix operations: addition, multiplication, transpose

Matrix inverses and elementary matrices

LU factorization

TEXTBOOKS

Introduction to Linear Algebra

Gilbert Strang · Wellesley-Cambridge, 6th Ed.

★ The MIT 18.06 textbook. Clear geometric intuition with full rigor.

Core

Linear Algebra Done Right

Sheldon Axler · Springer, 4th Ed.

Abstract approach — excellent once fundamentals are comfortable.

Intermediate

Vector Spaces & Linear Transformations

MODULE 04A

WEEKS 10–11

Vector Spaces

Moving beyond coordinates to abstract structure. What makes a collection of objects a "vector space" — and why this abstraction is so powerful.

Vector spaces and subspaces

Linear independence

Span, basis, and dimension

Null space, column space, row space

Rank-nullity theorem

MODULE 04B

WEEKS 11–12

Linear Transformations

Every matrix represents a transformation. Understanding this connection is the conceptual breakthrough of linear algebra — geometry and algebra become the same language.

Linear maps and their matrix representations

Kernel and image of a transformation

Change of basis

Composition of transformations

Geometric interpretation (rotations, reflections, projections)

TEXTBOOKS FOR MODULE 04

Introduction to Linear Algebra

Gilbert Strang · Chapters 3–7

Strang's "four fundamental subspaces" framework makes everything click.

Core

Linear Algebra and Its Applications

David C. Lay · Pearson, 6th Ed.

More application-focused alternative. Great worked examples.

Alternative

Eigenvalues, Determinants & Decompositions

MODULE 05A

WEEKS 13–14

Determinants & Eigenvalues

Determinants measure how a transformation scales volume. Eigenvalues reveal the "natural directions" of a matrix — where the transformation simply stretches or shrinks.

Determinant: cofactor expansion, properties

Eigenvalues and eigenvectors

Characteristic polynomial

Diagonalization

Complex eigenvalues

MODULE 05B

WEEKS 14–15

Orthogonality & Least Squares

Inner products, orthogonal projections, and the least squares method — the bridge between pure linear algebra and practical data fitting, signal processing, and robotics.

Inner product spaces and norms

Orthogonal and orthonormal bases

Gram-Schmidt process

QR decomposition

Least squares approximation

Intro to Singular Value Decomposition (SVD)

TEXTBOOKS FOR MODULE 05

Introduction to Linear Algebra

Gilbert Strang · Chapters 6–8

★ Eigenvalues, SVD, and applications. The culmination of 18.06.

Core

Linear Algebra Done Right

Sheldon Axler · Chapters 5–7

Determinant-free approach to eigenvalues. A different perspective worth reading.

Supplemental

Multivariable Calculus Essentials

MODULE 06A

WEEKS 16–17

Partial Derivatives & Gradients

Extending differentiation to functions of multiple variables. The gradient vector points in the direction of steepest ascent — a concept central to optimization and machine learning.

Functions of several variables, level curves

Partial derivatives

Gradient vector and directional derivatives

Chain rule for multivariable functions

Optimization: critical points, Hessian matrix

Lagrange multipliers

MODULE 06B

WEEKS 17–18

Multiple Integrals & Vector Calculus

Double and triple integrals over regions, then the key theorems of vector calculus — connecting line integrals, surface integrals, and the classical theorems.

Double integrals and iterated integrals

Triple integrals and change of variables

Polar, cylindrical, spherical coordinates

Line integrals and Green's Theorem

Divergence and curl

Stokes' Theorem and Divergence Theorem (overview)

TEXTBOOKS FOR MODULE 06

Calculus: Early Transcendentals

James Stewart · Chapters 14–16

Multivariable chapters. Clear diagrams and well-sequenced exercises.

Core

Vector Calculus, Linear Algebra, and Differential Forms

Hubbard & Hubbard · Matrix Editions, 5th Ed.

Unifies calculus and linear algebra beautifully. Ideal for deeper understanding.

Advanced

—

18-Week Schedule

Study Timeline

~8–12 HRS / WEEK RECOMMENDED

WK 1

Limits: intuition, one-sided limits, squeeze theorem MOD 01A

WK 2

Epsilon-delta, continuity, limit definition of derivative MOD 01A/B

WK 3

Differentiation rules, chain rule, implicit differentiation MOD 01B

WK 4

Mean Value Theorem, curve sketching, extrema MOD 02A

WK 5

Optimization, related rates, antiderivatives MOD 02A/B

WK 6

Fundamental Theorem of Calculus, u-sub, integration by parts MOD 02B

WK 7

Vectors, dot/cross product, systems of linear equations MOD 03

WK 8

Gaussian elimination, row echelon form, matrix operations MOD 03

WK 9

Matrix inverses, elementary matrices, LU factorization MOD 03

WK 10

Vector spaces, subspaces, linear independence MOD 04A

WK 11

Basis, dimension, null/column/row spaces, rank MOD 04A/B

WK 12

Linear transformations, change of basis, geometric interpretation MOD 04B

WK 13

Determinants, eigenvalues, eigenvectors MOD 05A

WK 14

Diagonalization, complex eigenvalues, orthogonality MOD 05A/B

WK 15

Gram-Schmidt, QR decomposition, least squares, intro SVD MOD 05B

WK 16

Partial derivatives, gradient, directional derivatives MOD 06A

WK 17

Hessian, multivariable optimization, Lagrange multipliers MOD 06A/B

WK 18

Double/triple integrals, Green's Theorem, divergence & curl MOD 06B

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Free Online Resources

MIT OpenCourseWare

18.01 Single Variable Calculus

Full video lectures by Prof. David Jerison, problem sets, and exams. Covers modules 1–2 completely. Free at ocw.mit.edu

MIT OpenCourseWare

18.02 Multivariable Calculus

Prof. Denis Auroux's full lecture series. Partial derivatives through vector calculus. Covers module 6. Free at ocw.mit.edu

MIT OpenCourseWare

18.06 Linear Algebra

Prof. Gilbert Strang's legendary lecture series. 34 video lectures covering modules 3–5. Free at ocw.mit.edu

YouTube · 3Blue1Brown

Essence of Linear Algebra

16 short visual videos building geometric intuition for vectors, transformations, eigenvalues, and more. Watch alongside modules 3–5.

YouTube · 3Blue1Brown

Essence of Calculus

12 beautifully animated videos covering derivatives, integrals, and the Fundamental Theorem. Ideal companion for modules 1–2.

Interactive · Free

Khan Academy: Calculus & Linear Algebra

Step-by-step practice problems with instant feedback. Covers all 6 modules at a beginner-friendly pace. Free at khanacademy.org

CALCULUS &LINEARALGEBRA

CALCULUS &
LINEAR
ALGEBRA