یادداشت‌های درسی برای ریاضی ۱۱۵A (جبر خطی)

These are lecture notes by Tao for a course in UCLA 2002, on the overview Tao writes: «This course is an introduction to Linear algebra. [...] Linear algebra is the study of the algebraic properties of linear transformations (and matrices). Algebra is concerned with how to manipulate symbolic combinations of objects, and how to equate one such combination with another; e.g. how to simplify an expression such as (x − 3)(x + 5). In linear algebra we shall manipulate not just scalars, but also vectors, vector spaces, matrices, and linear transformations. «These manipulations will include familiar operations such as addition, multiplication, and reciprocal (multiplicative inverse), but also new operations such as span, dimension, transpose, determinant, trace, eigenvalue, eigenvector, and characteristic polynomial. [Algebra is distinct from other branches of mathematics such as combinatorics (which is more concerned with counting objects than equating them) or analysis (which is more concerned with estimating and approximating objects, and obtaining qualitative rather than quantitative properties).]» Week 1 (Sections: 1.1-1.6) Overview of course Overview of course What is a vector? What is a vector space? Definition of a vector space (Not very important) remarks Examples of vector spaces Non-vector spaces Vector arithmetic Vector subspaces Linear combinations Spanning sets Linear dependence and independence Bases Week 2 (Sections: 1.6-2.1) Review of bases Examples of bases Rigorous treatment of bases Dimension Subspaces and dimension Lagrange interpolation Linear transformations Week 3 (Sections: 2.1-2.3) Review of linear transformations Null spaces and nullity Range and rank The dimension theorem Linear transformations and bases Co-ordinate bases The matrix representation of linear transformations Things to do with linear transformations Addition and multiplication of matrices Week 4 (Sections: 2.3-2.4) A quick review of matrices Co-ordinate matrices and composition Comparison between linear transformations and matrices Matrices as linear transformations. Invertible linear transformations Invertible linear transformations and invertible matrices Week 5 (Sections: 1.1-2.5) Changing the basis Co-ordinate change and matrices Common sources of confusion Week 6 (Sections: 3.1-5.1) Review: Row and column operations on matrices Rank of a matrix Determinants Geometric interpretation of determinants (optional) Week 7 (Sections: 4.5, 5.1-5.2) Cramer’s rule Diagonal matrices Eigenvalues and eigenvectors Computing eigenvalues Week 8 (Sections: 5.2, 6.1) Characteristic polynomials Tests for diagonalizability Inner product spaces Inner products and length Week 9 (Sections: 6.1-6.2) Orthogonality Orthonormal bases The Gram-Schmidt orthogonalization process. Orthogonal complements Week 10 (Sections: 3.1-5.1) Linear functionals Adjoints Normal operators Self-adjoint operators Assignment 1 (Sections: 1.1-1.6) Assignment 2 (Sections: 1.6-2.1) Assignment 3 (Sections: 2.1-2.3) Assignment 4 (Sections: 2.3-2.4) Midterm Assignment 5 (Sections: 2.4-2.5) Assignment 6 (Sections: 3.1-3.2, 4.1-4.4) Assignment 7 (Sections: 4.4, 5.1-5.2) Assignment 8 (Sections: 5.2, 6.2-6.3) Final Examination "Lecture notes from the Math 115A undergraduate course in linear algebra given in Fall 2002 at UCLA."