Math V: Advanced Linear Algebra¶
Year 2 · Block 1a · Mathematics
Eigenvalues lead into linear difference and differential equations and their stability, then into orthogonality, the four fundamental subspaces, the singular value decomposition, generalized inverses and least squares.
- 7 weeks
- 40 concept pages
Practise¶
Weeks¶
1Eigenvalues and linear maps8 concepts
- Determinant Math II: Linear Algebra
- Eigenvalue and Eigenvector Math II: Linear Algebra
- Identity Matrix Math II: Linear Algebra
- Matrix Product Math II: Linear Algebra
- Matrix Representation of a Linear Map
- Characteristic Polynomial Math II: Linear Algebra
- Complex Conjugate Pair of Eigenvalues
- Inverse and Singular Matrices Math II: Linear Algebra
Review week 1 in Concept Finder 8 concepts
2Linear difference equations9 concepts
- Difference Equation
- First-Order Linear Difference Equation
- Second-Order Linear Difference Equation
- System of Linear Difference Equations
- Characteristic Polynomial for a Difference Equation
- Higher-Order Linear Difference Equation
- Stability of a Scalar Difference Equation
- Stability of a System of Linear Difference Equations
- Basis of Solutions of a Difference Equation
Review week 2 in Concept Finder 9 concepts
3First-order differential equations5 concepts
- Differential Equation
- First-Order Differential Equation
- Integrating Factor
- Equilibrium of a Differential Equation
- Stability of an Equilibrium
Review week 3 in Concept Finder 5 concepts
4Higher-order equations and linear systems8 concepts
- Matrix Exponential
- Second-Order Differential Equation
- System of Differential Equations
- Characteristic Equation for a Differential Equation
- Linear Vector ODE
- Particular Solution of an Inhomogeneous Linear ODE
- General Solution of a Linear ODE
- Stability of Solutions of Higher-Order ODEs
Review week 4 in Concept Finder 8 concepts
5Orthogonality and the four subspaces18 concepts
- Euclidean Norm Math II: Linear Algebra
- Inner Product Math II: Linear Algebra
- Linear Combination Math II: Linear Algebra
- Linear Dependence and Independence Math II: Linear Algebra
- Row Echelon and Reduced Row Echelon Form Math II: Linear Algebra
- Vector Space Math II: Linear Algebra
- Linear Span Math II: Linear Algebra
- Subspace Math II: Linear Algebra
- Basis Math II: Linear Algebra
- Dimension Math II: Linear Algebra
- Kernel Math II: Linear Algebra
- Orthogonal Complement
- Orthogonal Projection
- Range of a Linear Map Math II: Linear Algebra
- Rank Math II: Linear Algebra
- Row Space
- Affine Subspace
- Fundamental Theorem of Linear Algebra
Review week 5 in Concept Finder 18 concepts
6Orthonormal bases and change of basis7 concepts
- Block Matrix Math II: Linear Algebra
- Change of Basis
- Gram-Schmidt Algorithm
- Kronecker Product
- Orthonormal Basis
- Similarity Transformation
- Orthogonal Matrix
Review week 6 in Concept Finder 7 concepts
7Singular values, generalized inverses and least squares6 concepts
- Least Squares
- Left and Right Inverses Math IV: Multivariate Calculus
- Singular Value
- Singular Value Decomposition
- Generalized Inverse
- Moore-Penrose Inverse
Review week 7 in Concept Finder 6 concepts
Concepts appear in the week that introduces them, foundations first. A course name after a concept marks a page from another course used that week.
More about this course¶
Before you startEarlier concepts these pages use most. If a step feels unfamiliar, revisit them first.4 courses
- Eigenvalue and Eigenvector
- Matrix Product
- Euclidean Norm
- Basis
- Diagonalization
- Inner Product
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+35 moreShow fewer
- Inverse and Singular Matrices
- Characteristic Polynomial
- Linear Dependence and Independence
- Determinant
- Kernel
- Range of a Linear Map
- Rank
- Subspace
- Vector Space
- Block Matrix
- Dimension
- Jordan Normal Form
- Linear Span
- Row Echelon and Reduced Row Echelon Form
- Identity Matrix
- Linear Combination
- Orthogonal Vectors
- Symmetric Matrix
- Transpose
- Complex Absolute Value
- Coordinate
- Euler's Formula
- Homogeneous and Inhomogeneous Systems
- Root of a Polynomial
- Complex Conjugate
- Complex Conjugate Root Theorem
- Complex Number
- Definite Matrix
- Direct Sum
- Eigenbasis
- Elementary Row Operation
- Fundamental Theorem of Linear Maps
- Linear Map
- Matrix Scalar Multiplication
- System of Linear Equations
Also: Taylor Series (Math III: Analysis)
AchievementsCourse challenges and how many learners hold them
How to read this courseNotation, shared pages and conventions
These notes introduce course-specific concepts. Linear-algebra material already taught in Math II (and the Math I calculus used for differential equations) has one canonical page; it is linked under Before you start or shown in the week that revisits it, rather than repeated. Week numbers are counted within this course and follow the weekly slide decks; Weeks 3 and 4 share one deck, as do Weeks 6 and 7. Concepts appear in the week where they are first introduced and are reused by links afterwards.
SourcesCourse material these pages were checked against
- Math V: Advanced Linear Algebra · Weekly lecture slides (Weeks 1–7)
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: M5, Math 5
Last updated: 26 October 2026 00:09