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
For teachers and tutorsPrepare the connections for your next explanation
Choose a concept for your next explanation. Review its recorded foundations and later uses, then check the concept page's sources against your own course material.
The workbench uses published prerequisite links. It describes concept relationships; class performance and teaching coverage need separate evidence.
Aliases: M2, Math 2
Last updated: 26 October 2026 00:09