Math II: Linear Algebra¶
Year 1 · Block 1b · Mathematics
Connect vector spaces and linear maps with matrices, geometry and eigenvalues.
- 6 weeks
- 139 concept pages
Practise¶
Weeks¶
1Sets, algebraic structures and complex numbers53 concepts
- Cardinality
- Cardinality of the Continuum
- Cartesian Product
- Countable and Uncountable Sets
- Equivalence Relation
- Imaginary Unit
- Permutation
- Polar Coordinate System
- Power Set
- Root of a Polynomial
- Aleph Null
- Binary Operation
- Cantor's Diagonal Argument
- Cantor's Theorem
- Commutativity
- Complex Number
- Complex Trigonometric Functions
- Continuum Hypothesis
- Distributivity
- Fundamental Theorem of Algebra
- Identity Element
- Inverse Element
- 2-Argument Arctangent
- Algebraic Structure
- Associativity
- Cartesian Representation of Complex Numbers
- Complex Absolute Value
- Complex Addition
- Complex Conjugate
- Complex Conjugate Root Theorem
- Complex Exponential Function
- Complex Multiplication
- Complex Negation
- Complex Reciprocal
- Complex Subtraction
- Group
- Polar Representation of Complex Numbers
- Semigroup
- Symmetric Group
- Abelian Group
- Argument of a Complex Number
- Complex Division
- Complex Logarithm
- De Moivre's Formula
- Euler's Formula
- Euler's Identity
- Monoid
- Ring
- Complex Exponentiation
- Field
- Polynomial Long Division
- Algebraically Closed Field
- Polynomial Greatest Common Divisor
Review week 1 in Concept Finder 53 concepts
2Vector spaces, span and bases19 concepts
- Field
- Vector Space
- Coordinate
- Finite-Dimensional Vector Space
- Scalar Multiplication
- Vector
- Vector Addition
- Zero Vector
- Linear Combination
- Linear Dependence and Independence
- Linear Span
- Origin
- Real Coordinate Space
- Spanning List
- Subspace
- Sum of Subspaces
- Basis
- Dimension
- Direct Sum
Review week 2 in Concept Finder 19 concepts
3Matrices, linear systems and linear maps20 concepts
- Linear Map
- Linear Operator
- Matrix
- Matrix Addition
- Matrix Product
- Matrix Scalar Multiplication
- Pivot
- Product of Linear Maps
- Row Echelon and Reduced Row Echelon Form
- Transpose
- Vector
- Elementary Row Operation
- Identity Matrix
- Symmetric Matrix
- System of Linear Equations
- Augmented Matrix
- Gauss-Jordan Elimination
- Solution Set
- Standard Form of a Linear System
- Equivalent Systems of Equations
Review week 3 in Concept Finder 20 concepts
4Determinants, kernel, range and rank13 concepts
- Determinant
- Homogeneous and Inhomogeneous Systems
- Inverse and Singular Matrices
- Kernel
- Matrix Diagonals
- Range of a Linear Map
- Rank
- Sarrus' Rule
- System of Linear Equations
- Cofactor Expansion
- Cramer's Rule
- Fundamental Theorem of Linear Maps
- Adjugate Matrix
Review week 4 in Concept Finder 13 concepts
5Inner products and geometry21 concepts
- Block Matrix
- Diagonal Matrix
- Euclidean Norm
- Euclidean Space
- Idempotent Matrix
- Inner Product
- Orthogonal Vectors
- Parametric Representations of Lines and Planes
- Permutation Matrix
- Plane
- Skew Lines
- Triangular Matrix
- Angle Between Vectors
- Cauchy-Schwarz Inequality
- Euclidean Distance
- LU Decomposition
- Normal Vector
- Point-Normal Equation of a Plane
- Triangle Inequality for Vector Norms
- Distance Between a Point and a Line
- Hyperplane
Review week 5 in Concept Finder 21 concepts
6Eigenvalues, diagonalization and quadratic forms16 concepts
- Characteristic Polynomial
- Eigenvalue and Eigenvector
- Eigenvalue-Root Theorem
- Generalized Eigenvector
- Invariant Subspace
- Jordan Blocks and Chains
- Jordan Normal Form
- Quadratic Form
- Trace
- Algebraic and Geometric Multiplicity
- Definite Matrix
- Definiteness and Eigenvalues
- Distinct Eigenvalues Theorem
- Eigenbasis
- Diagonalizable Matrix
- Diagonalization
Review week 6 in Concept Finder 16 concepts
Concepts appear in the week that introduces them, foundations first. Week titles summarise the concepts tagged to each week; a course name after a concept marks a page from another course used that week.
Set theory and cardinalitySet-theory recall, comparing sizes, and Cantor's results.10 concepts
Set-theory recall 3
Comparing sizes 4
Cantor's results 3
Algebraic and complex foundationsAlgebraic structures, complex numbers, polynomials, and formulas used later.44 concepts
Algebraic language 8
Algebraic structures 9
Complex-number basics 4
Complex arithmetic 6
Complex geometry 5
Exponential, trig, and logarithm 6
Vector spacesLinear combinations, span, independence, bases, and dimension.18 concepts
Spaces and subspaces 9
Span and combinations 3
Independence and bases 2
Dimension and coordinates 4
Matrices, systems, and linear mapsMatrix operations, linear systems, row reduction, and maps.26 concepts
Matrix essentials 7
Special matrices 6
Linear systems 5
Row reduction 5
Linear maps 3
Rank, kernel, inverse matrices, and determinantsInvertibility, rank, determinant methods, and solution structure.11 concepts
Kernel, range, and rank 4
Invertibility and systems 2
Determinants 5
Inner products and geometryLengths, angles, orthogonality, projections, lines, and planes.16 concepts
Lengths and angles 6
Orthogonality and projections 3
Eigenvalues, diagonalisation, and quadratic formsSpectral ideas, diagonalisation, Jordan form, and definiteness.15 concepts
Eigenvalue basics 6
Diagonalisation 6
Quadratic forms 3
More about this course¶
Before you startEarlier concepts these pages use most. If a step feels unfamiliar, revisit them first.1 course
Where it leadsLater courses on the wiki that build on these concepts.7 courses
- Eigenvalue and Eigenvector
- Matrix Product
- Euclidean Norm
- Basis
- Diagonalization
- Inner Product
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+33 moreShow fewer
- Inverse and Singular Matrices
- Orthogonal Vectors
- Characteristic Polynomial
- Linear Dependence and Independence
- Symmetric Matrix
- Transpose
- Complex Absolute Value
- Coordinate
- Determinant
- Euler's Formula
- Homogeneous and Inhomogeneous Systems
- Kernel
- Range of a Linear Map
- Rank
- Root of a Polynomial
- Subspace
- Vector Space
- Block Matrix
- Complex Conjugate
- Complex Conjugate Root Theorem
- Complex Number
- Definite Matrix
- Dimension
- Direct Sum
- Eigenbasis
- Elementary Row Operation
- Fundamental Theorem of Linear Maps
- Jordan Normal Form
- Linear Map
- Linear Span
- Matrix Scalar Multiplication
- Row Echelon and Reduced Row Echelon Form
- System of Linear Equations
Also: Risk Insurance (via Matrix Product), Stochastic Models (via Matrix Product), Game Theory (via Cartesian Product)
AchievementsCourse challenges and how many learners hold them
Aliases: M2, Math 2
Last updated: 26 October 2026 00:09