OverviewWhat Is Linear Algebra?
Linear algebra studies linear equations, the spaces they live in, and the transformations that act on those spaces: all through the lens of vectors and matrices.
At its core, linear algebra asks a deceptively simple question: given a system of linear equations, does a solution exist, how many solutions are there, and how do we find them efficiently? The answer turns out to involve a deep interplay between geometry (vectors as arrows in space), algebra (matrices as compact representations of systems), and abstraction (vector spaces as the general framework that makes everything work).
Mastery of linear algebra is a prerequisite for multivariable calculus, differential equations, statistics, machine learning, quantum mechanics, and computer graphics. Every time a neural network trains, a simulation runs, or a 3D model rotates on screen, linear algebra is the engine underneath.
Systems of equations
Ax = b
The central problem: find x given A and b
Eigenvalue equation
Av = λv
The vector v is only scaled, not rotated
Determinant
det(A) = ad − bc
Measures area/volume scaling factor
Diagonalisation
A = PDP⁻¹
Unlocks efficient computation of Aⁿ
Where to Start
If you're completely new to linear algebra, begin with Vectors and Vector Spaces — it introduces all the geometric intuition you'll need for everything else. If you're reviewing for an exam, jump straight to the topic you need using the topic grid below.
All TopicsComplete Linear Algebra Course
Each topic page contains a full written explanation (1,000–2,000 words), multiple worked examples escalating in difficulty, and a 10-question interactive quiz with full solutions.
Topic 01
Vectors & Vector Spaces
Geometric and algebraic vectors in ℝⁿ. Vector addition and scalar multiplication. The eight vector space axioms. Dot product, cross product, magnitude, and angle. Subspaces, span, linear independence, basis, and dimension.
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Topic 02
Matrices — Introduction
Matrix notation, size, and entries. Types of matrices: zero, identity, square, diagonal, symmetric, triangular. Matrix addition and scalar multiplication. The transpose and its properties. Setting up augmented matrices.
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Topic 03
Matrix Multiplication
The conformability rule: (m×k)(k×n) = m×n. Computing (AB)ᵢⱼ as a dot product of row i and column j. Why AB ≠ BA in general. The identity matrix. Matrix powers. The transpose reversal rule (AB)ᵀ = BᵀAᵀ. Block multiplication.
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Topic 04
Determinants
The 2×2 determinant: ad − bc. Cofactor expansion for 3×3 and larger matrices. Key properties: det(AB) = det(A)det(B), det(Aᵀ) = det(A). Geometric interpretation as area and volume scaling. Singular matrices and Cramer's Rule.
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Topic 05
Inverse Matrices
Definition: AA⁻¹ = I. The 2×2 inverse formula. Gauss-Jordan elimination via the augmented matrix [A|I] → [I|A⁻¹]. Conditions for invertibility. Properties: (AB)⁻¹ = B⁻¹A⁻¹. Solving Ax = b via A⁻¹.
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Topic 06
Systems of Linear Equations
Writing systems as Ax = b. The three cases: unique solution, infinitely many, no solution. Geometric interpretation in 2D and 3D. Substitution and elimination methods. The relationship between invertibility of A and the solution set.
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Topic 07
Row Reduction & Echelon Form
Augmented matrix [A|b]. The three elementary row operations. Row Echelon Form (REF) and Reduced Row Echelon Form (RREF). Pivot positions, free variables, and the solution space. Gaussian and Gauss-Jordan elimination with full worked examples.
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Topic 08
Eigenvalues & Eigenvectors
Definition: Av = λv — the vector is only stretched, not rotated. The characteristic polynomial det(A − λI) = 0. Finding eigenvalues and their corresponding eigenspaces. The trace-eigenvalue and determinant-eigenvalue relationships. Complex eigenvalues.
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Topic 09
Diagonalisation
When is a matrix diagonalisable? Building A = PDP⁻¹ from eigenvectors and eigenvalues. Computing Aⁿ efficiently via diagonalisation. Symmetric matrices are always diagonalisable (the spectral theorem). Applications in differential equations and Markov chains.
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Topic 10
Linear Transformations
Functions T: V → W satisfying T(u+v) = T(u)+T(v) and T(cv) = cT(v). Matrix representation of linear transformations. The kernel (null space) and image (range). The Rank-Nullity Theorem: dim(ker T) + dim(im T) = dim(V). Composition of transformations.
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Guided PathRecommended Study Order
Linear algebra topics build on each other. Follow this order on a first pass: you'll rarely need to skip ahead.
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1Vectors & Vector Spaces The geometric foundation. Learn what vectors are, how they combine, and what a vector space means. Everything else assumes this. Start here →
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2Matrices Introduction How to represent data and linear maps as rectangular arrays of numbers. Addition, transpose, and matrix types. Read →
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3Matrix Multiplication The dot-product definition of (AB)ᵢⱼ, why order matters, and the identity matrix. The gateway to everything downstream. Read →
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4Systems of Linear Equations + Row Reduction Study these two topics together. Row reduction is the algorithm; systems are what you're solving. Master Gaussian elimination. Systems → Row Reduction →
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5Determinants A single number summarising key information about a square matrix. Ties together invertibility, area/volume, and eigenvalues. Read →
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6Inverse Matrices When does A⁻¹ exist? How to compute it via Gauss-Jordan. The connection between det(A) ≠ 0 and invertibility. Read →
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7Eigenvalues & Eigenvectors The characteristic polynomial, eigenspaces, and the geometric meaning of "only stretching". Requires determinants and row reduction. Read →
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8Diagonalisation Assembling eigenvalues and eigenvectors into A = PDP⁻¹. Opens up efficient computation of matrix powers and ODEs. Read →
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9Linear Transformations The abstract framework that unifies everything: matrices as functions between vector spaces. Kernel, image, rank-nullity. Read →
Quick ReferenceEssential Formulas at a Glance
A condensed reference for the most-used results across the course. Each formula links to its full derivation and worked examples.
| Result | Formula | Where to learn it |
|---|---|---|
| Dot product | u · v = Σ uᵢvᵢ = |u||v|cosθ | Vectors |
| Orthogonality condition | u ⊥ v ⟺ u · v = 0 | Vectors |
| Matrix product entry | (AB)ᵢⱼ = row i of A · col j of B | Matrix Multiplication |
| Transpose of product | (AB)ᵀ = BᵀAᵀ | Matrix Multiplication |
| 2×2 determinant | det[[a,b],[c,d]] = ad − bc | Determinants |
| Invertibility condition | A invertible ⟺ det(A) ≠ 0 | Inverse Matrices |
| 2×2 inverse | A⁻¹ = (1/det A)[[d,−b],[−c,a]] | Inverse Matrices |
| Characteristic polynomial | det(A − λI) = 0 | Eigenvalues |
| Eigenvalue equation | Av = λv | Eigenvalues |
| Diagonalisation | A = PDP⁻¹, Aⁿ = PDⁿP⁻¹ | Diagonalisation |
| Rank-Nullity Theorem | rank(T) + nullity(T) = dim(V) | Linear Transformations |
| Trace–eigenvalue relation | tr(A) = λ₁ + λ₂ + … + λₙ | Eigenvalues |
ContextPrerequisites & Applications
What you need before starting
- Comfortable with algebraic manipulation and solving equations
- Basic coordinate geometry, plotting points, lines in the plane
- Some exposure to functions and function notation
- Familiarity with summation notation (Σ) is helpful but not essential
- Calculus is not required for the core linear algebra topics
What linear algebra unlocks
- Multivariable calculus, gradient, Jacobian, Hessian all use matrices
- Differential equations: systems of ODEs solved via eigenvalues
- Machine learning, neural networks, PCA, SVD are pure linear algebra
- Quantum mechanics, state vectors and operators live in Hilbert spaces
- Computer graphics: 3D rotations, projections, and transformations
- Statistics: covariance matrices, regression, dimensionality reduction
FAQCommon Questions
Is linear algebra harder than calculus?
Linear algebra and calculus require different kinds of thinking. Calculus is primarily about rates of change and requires comfort with limits and continuous reasoning. Linear algebra is about structure, systems, and transformation — it requires comfort with abstraction and careful logical argument. Most students find the first half of linear algebra (computation with matrices and row reduction) easier than calculus, while the second half (abstract vector spaces, eigentheory) can feel harder. Both require practice with worked examples.
Do I need to know calculus before studying linear algebra?
No. The core topics of linear algebra — vectors, matrices, row reduction, determinants, eigenvalues — are self-contained and do not require any calculus. You will encounter calculus when you apply linear algebra (for example, in differential equations or multivariable calculus), but you can learn the entire subject without it. Many universities teach linear algebra in the first year alongside or before calculus.
What is the most important topic in linear algebra?
If forced to choose one, most working mathematicians and engineers would say eigenvalues and eigenvectors. They appear in almost every serious application: solving differential equations, principal component analysis in statistics, Google's PageRank algorithm, the quantum harmonic oscillator, and the stability analysis of dynamical systems. Understanding eigentheory deeply is what separates competent linear algebra from true fluency in the subject. That said, you cannot reach eigenvalues without first mastering matrix multiplication, row reduction, and determinants.
Why can't you divide by a matrix?
Division by a number x is equivalent to multiplying by 1/x. For matrices, the equivalent of "1/A" is the inverse matrix A⁻¹, and it exists only when A is square and has a non-zero determinant. When the determinant is zero (A is singular), no inverse exists — analogous to dividing by zero in ordinary arithmetic. Even when A⁻¹ exists, because AB ≠ BA in general, you must be careful about whether you are "dividing from the left" (multiplying by A⁻¹ on the left) or "from the right" — two different operations with two different results.
What is the difference between a matrix and a linear transformation?
A linear transformation is an abstract function T: V → W between vector spaces that satisfies two rules: T(u+v) = T(u)+T(v) and T(cu) = cT(u). A matrix is a concrete rectangular array of numbers. The deep result connecting them is that every linear transformation between finite-dimensional vector spaces can be represented as a matrix (once you choose bases for V and W), and every matrix represents a linear transformation. They are two different descriptions of the same underlying object — the abstract map and its concrete numerical representation.
How is linear algebra used in machine learning?
Linear algebra is the language of modern machine learning. Neural network weights are stored in matrices; forward propagation is matrix multiplication; backpropagation involves transposes and Jacobians. Principal Component Analysis (PCA) — used to compress data — is exactly the eigenvalue decomposition of a covariance matrix. Singular Value Decomposition (SVD), an extension of diagonalisation, underlies recommendation systems and natural language processing. The training of any linear regression model reduces to solving a linear system Ax = b. Understanding linear algebra deeply makes machine learning algorithms transparent rather than black-box.