18.700
Linear Algebra
Vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. More emphasis on theory and proofs than in 18.06 .
Fall 2013
Topics
- Definition of vector spaces, properties
- Subspaces, sums and direct sums
- Span, independence, bases
- Bases and dimension
- Linear maps, null space / range
- matrices, invertibility
- Finite fields. Systems of equations.
- Gaussian elimination
- Counting over F p . Invariant subspaces.
- Finding eigenvectors
- Upper triangular and diagonal matrices
- Eigen vectors over R and F p
- Inner products, Gram-Schmidt
- Orthogonal projection, minimization
- Adjoint, self-adjoint, normal
- Spectral theorem
- Positive operators
- Isometries, polar decomposition
- Generalized eigenspaces
- Generalized eigenspace decomposition
- Characteristic polynomial
- Determinant
- Trace, canonical commutation relations
Readings
- Most of the reading assignments below refer to the textbook:
- LEC #: 1 · TOPICS: Definition of vector spaces, properties · READINGS: [Axler] Chapter 1, pp. 2–12.
- LEC #: 2 · TOPICS: Subspaces, sums and direct sums · READINGS: [Axler] Chapter 1, pp. 13–18.
- LEC #: 3 · TOPICS: Span, independence, bases · READINGS: [Axler] Chapter 2, pp. 21–31.
- LEC #: 4 · TOPICS: Bases and dimension · READINGS: [Axler] Chapter 2, pp. 31–34.
- LEC #: 5 · TOPICS: Linear maps, null space / range · READINGS: [Axler] Chapter 3, pp. 38–47.
- LEC #: 6 · TOPICS: Matrices, invertibility · READINGS: [Axler] Chapter 3, pp. 48–58.
- LEC #: 11 · TOPICS: Finding eigenvectors · READINGS: [Axler] Chapter 5, pp. 80–81.
- LEC #: 12 · TOPICS: Upper triangular and diagonal matrices · READINGS: [Axler] Chapter 5, pp. 81–90.
- LEC #: 16 · TOPICS: Orthogonal projection, minimization · READINGS: [Axler] Chapter 6, pp. 111–16.
- LEC #: 17 · TOPICS: Adjoint, self-adjoint, normal · READINGS: [Axler] Chapter 7, pp. 127–32.
- LEC #: 19 · TOPICS: Positive operators · READINGS: [Axler] Chapter 7, pp. 144–47.
- LEC #: 20 · TOPICS: Isometries, polar decomposition · READINGS: [Axler] Chapter 7, pp. 147–57.
- LEC #: 24 · TOPICS: Characteristic polynomial · READINGS: [Axler] Chapter 8, pp. 168–73.
- LEC #: 25 · TOPICS: Determinant · READINGS: [Axler] Chapter 10.
- LEC #: 26 · TOPICS: Trace, canonical commutation relations · READINGS: [Axler] Chapter 10.
Prerequisites & course information
Prereq: Calculus II (GIR) U (Fall) 3-0-9 units. REST Credit cannot also be received for 6.C06[J] , 18.06 , 18.C06[J] , CC.1806 , ES.1806
Sources
- Course catalogue ↗2026–2027
- MIT OpenCourseWare ↗Fall 2013
- MIT OCW readings ↗
- MIT OCW calendar ↗
- MIT OCW syllabus ↗
- Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.