Class

lecture | tutorial | |

Time | Wed, Fri | Tue |

13:30 - 14:50 | 20:00 - 20:50 | |

Venue | 2504 | 2406 |

Instructor | YAN Min | |

Office | 3487 | |

Phone | 23587442 |

Prerequisite

Basic geometrical imagination, solving systems of linear equations, basic knowledge of polynomials.

Topics

Vector Space

linear combination, span, linear independence, system of linear equations, row echelon form, basis, coordinate, dimension

Linear Transformation

linear transformation, composition, matrix operation, one-to-one, onto, isomorphism, matrix of linear transformation, change of basis

Subspace

span, range, rank, kernel, sum, direct sum, projection, blocks of linear transformation, quotient space, direct summand

Inner Product

dot product, inner product, adjoint, orthogonality, orthonormal basis, isometry, orthogonal matrix, Gram-Schmidt, orthogonal complement, orthogonal projection, complementarity principal

Determinant

determinant of square matrix, permutation, row and column operation on determinant, cofactor expansion, determinant of linear operator, orientation, volume

Advanced Vector Space

complex number, complex inner product, complex vs real structure, module over ring, abelian group, polynomial

Eigenvalue and Eigenvector

eigenspace, characteristic polynomial, diagonalization, normal operator, hermitian operator, unitary operator

Spectral Theory

invariants of linear operator, Cayley-Hamilton Theorem, algebraic and geometric multiplicity, invariant subspace, Jordan canonical form, minimal polynomial, other canonical forms

Tensor

bilinear function, quadratic form, signature, positive definite, multilinear function, tensor of vector space, exterior algebra

Homework

No hard copies of the homework will be distributed. Please check my web page for the latest assignments. Usually you have one week to do the homework. The answer will be put on my web page after the homework is collected. *Absolutely no late homework will be accepted*.

Exam and Grade

Homework 20%, midterm 30%, final 50%.

Note

The syllabus is subject to change as circumstances arise.