Linear algebra
Undergraduate · Math
Syllabus focus
Standard syllabus · STEM / applied · Theoretical / proof-based
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Topics typically covered
Standard syllabus
Systems and matrix algebra
- Systems of linear equations; Gaussian elimination and row echelon form
- Matrix algebra: addition, multiplication, and properties
- Matrix inverses and the Invertible Matrix Theorem (statement)
- Determinants and their geometric interpretation (area/volume scaling)
Vector spaces
- Vectors in R^n; linear combinations and span
- Linear independence and dependence
- Subspaces; column space, null space, and bases
- Dimension and the rank-nullity theorem (statement)
Eigenvalues and orthogonality
- Eigenvalues and eigenvectors; diagonalization of square matrices
- Orthogonality, projections, and the Gram–Schmidt process
- Least squares solutions to inconsistent systems
- Inner product spaces and orthonormal bases (introduction)
STEM / applied
Computation and applications
- Computational linear algebra with technology (MATLAB, Python, or calculator)
- Linear transformations and their matrix representations
- Applications to computer graphics: rotations, scaling, and composition
- Least squares in data fitting and regression
Modeling
- Markov chains and steady-state vectors (introduction)
- Network flow and incidence matrices (optional applications)
- Numerical stability and conditioning (introduction)
- Singular value decomposition (overview at applied level)
Theoretical / proof-based
Vector space proofs
- Vector space axioms and subspace proofs
- Proofs of linear independence and basis theorems
- Rank-nullity theorem: proof and consequences
- Abstract vector spaces beyond R^n (polynomials, functions)
Maps and diagonalization
- Linear transformations: kernel, image, and isomorphism
- Change of basis and similarity of matrices
- Characteristic polynomials and algebraic multiplicity
- Diagonalization criteria and proofs
- Orthogonality proofs and the projection theorem
- Spectral theorem for symmetric matrices (statement and proof sketch)
Notes
Topics reflect common linear algebra syllabi at US colleges and universities. Applied/engineering sections emphasize computation; proof-based sections mirror “linear algebra for mathematics majors.”