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)
Aliases: M5, Math 5
Last updated: 26 October 2026 00:09