Standard syllabus
Linear algebra · Undergraduate · Math
Learning objectives from the Linear algebra syllabus, grouped by unit. Click an objective for study materials.
Topics typically covered
Click a topic for the full text and related unit practice.
Undergraduate Linear Algebra — scope drawn from Axler Linear Algebra Done Right (open PDF), Treil Linear Algebra Done Wrong, Beezer A First Course in Linear Algebra, and AIM-approved OER texts.
Vector spaces and bases
- Axioms of vector spaces — Axioms of vector spaces
- Subspaces over R (and C when needed) — Subspaces over R (and C when needed)
- Linear combinations and span — Linear combinations and span
- Linear independence — Linear independence
- Bases and dimension — Bases and dimension
- Coordinate representations — Coordinate representations
- Rank-nullity for linear maps (statement — Rank-nullity for linear maps (statement
- Proof outline) — Proof outline)
Linear maps and matrices
- Definition — Definition
- Examples of linear maps between vector spaces — Examples of linear maps between vector spaces
- Kernel, image, and injectivity — Kernel, image, and injectivity
- Surjectivity — Surjectivity
- Matrix of a linear map relative to ordered bases — Matrix of a linear map relative to ordered bases
- Change of basis and similar matrices — Change of basis and similar matrices
Determinants (theory)
- Axiomatic characterization of the determinant — Axiomatic characterization of the determinant
- Existence — Existence
- Uniqueness — Uniqueness
- Det(AB) = det(A)det(B) — Det(AB) = det(A)det(B)
- Invertibility criteria — Invertibility criteria
- Geometric interpretation via signed volume — Geometric interpretation via signed volume
Eigenvalues and diagonalization
- Eigenvalues and eigenvectors — Eigenvalues and eigenvectors
- Characteristic polynomials — Characteristic polynomials
- Invariant subspaces — Invariant subspaces
- Triangularization over C (intro) — Triangularization over C (intro)
- Diagonalizability criteria — Diagonalizability criteria; algebraic vs geometric multiplicity
- Applications to systems of linear ODEs (light touch) — Applications to systems of linear ODEs (light touch)
Inner products and the spectral theorem (intro)
- Inner products and norms — Inner products and norms
- Cauchy-Schwarz — Cauchy-Schwarz
- Orthogonal — Orthogonal
- Orthonormal bases — Orthonormal bases
- Orthogonal projections — Orthogonal projections
- Least squares in inner product language — Least squares in inner product language
- Spectral theorem for real symmetric / self-adjoint operators (statement and use) — Spectral theorem for real symmetric / self-adjoint operators (statement and use)
Learning objectives
Click an objective for study materials.
Vector spaces and bases
- Axioms of vector spaces — Axioms of vector spaces
- Subspaces over R (and C when needed) — Subspaces over R (and C when needed)
- Linear combinations and span — Linear combinations and span
- Linear independence — Linear independence
- Bases and dimension — Bases and dimension
- Coordinate representations — Coordinate representations
- Rank-nullity for linear maps (statement — Rank-nullity for linear maps (statement
- Proof outline) — Proof outline)
Linear maps and matrices
- Definition — Definition
- Examples of linear maps between vector spaces — Examples of linear maps between vector spaces
- Kernel, image, and injectivity — Kernel, image, and injectivity
- Surjectivity — Surjectivity
- Matrix of a linear map relative to ordered bases — Matrix of a linear map relative to ordered bases
- Change of basis and similar matrices — Change of basis and similar matrices
Determinants (theory)
- Axiomatic characterization of the determinant — Axiomatic characterization of the determinant
- Existence — Existence
- Uniqueness — Uniqueness
- Det(AB) = det(A)det(B) — Det(AB) = det(A)det(B)
- Invertibility criteria — Invertibility criteria
- Geometric interpretation via signed volume — Geometric interpretation via signed volume
Eigenvalues and diagonalization
- Eigenvalues and eigenvectors — Eigenvalues and eigenvectors
- Characteristic polynomials — Characteristic polynomials
- Invariant subspaces — Invariant subspaces
- Triangularization over C (intro) — Triangularization over C (intro)
- Diagonalizability criteria — Diagonalizability criteria; algebraic vs geometric multiplicity
- Applications to systems of linear ODEs (light touch) — Applications to systems of linear ODEs (light touch)
Inner products and the spectral theorem (intro)
- Inner products and norms — Inner products and norms
- Cauchy-Schwarz — Cauchy-Schwarz
- Orthogonal — Orthogonal
- Orthonormal bases — Orthonormal bases
- Orthogonal projections — Orthogonal projections
- Least squares in inner product language — Least squares in inner product language
- Spectral theorem for real symmetric / self-adjoint operators (stateme... — Spectral theorem for real symmetric / self-adjoint operators (statement and use)
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Vector spaces and bases
Axioms of vector spaces
Coming soon - Linear maps and matrices
Definition
Coming soon - Determinants (theory)
Axiomatic characterization of the determinant
Coming soon - Eigenvalues and diagonalization
Eigenvalues and eigenvectors
Coming soon - Inner products and the spectral theorem (intro)
Inner products and norms
Coming soon
Pricing calculator
Choose materials, tutoring, or both — or book a single session as needed. Customize your plan on the subscribe page.
$1,162 · Linear algebra · 18 tutoring hrs
Study guides, worksheets, reviews, practice tests, and answer keys for 1 class. 18 tutoring hours (1 hr / week · semester). Bundle discount applied vs buying separately. Pay in full via Zelle.