# 《〈几何与代数》课程简介(胡国权）

1. 向量代数与几何 ( 向量运算, 坐标系, 平面与直线 );

2. 常见曲面 (旋转面, 柱面和锥面, 二次曲面, 坐标变换与点变换 );

3. 基本代数结构 ( 群 环 域, 向量空间, 矩阵运算);

4. 线性代数基础 ( 线性方程组, 基与维数, 线性映射, 行列式, 特征值, 实二次型 )．

1. 多项式代数 (根, 整除与因式分解, 多元多项式);

2. 向量空间 ( 直和与商, 线性型与对偶, 双线性与二次型, 欧氏空间, 酉空间 );

3. 线性变换 ( 特征向量与不变子空间, 正交算子, 对称算子，正规算子, 若当标准形 );

4. 仿射与射影空间 ( 仿射空间, 仿射变换与运动, 二次曲面, 射影空间 )．

This is one of the fundamental courses for beginners in pure and applied mathematics , statistics etc.　The material of this course is an important part of the mathematical background that will be helpful to everyone regardless of the field of specialization. The objective of the course is to present the fundamentals of linear algebra and analytic geometry,　to help students learn the fundamental concepts, understand the basic principles,　learn to reason abstractly, develop their ability of solving problems,　and to provide a firm foundation that will enable the students to continue studies in mathematics and science.　The course is divided into two semesters.

The first semester course will cover the following topics:

1. Algebra of vectors and geometry ( operations of vectors, affine frame, planes and lines, Rn);

2. Surfaces (surfaces of revolution, cylinders , conics, quadrics, coordinate changes, isometries );

3. Basic algebraic structures ( groups, rings and fields, vector spaces , matrices);

4. An introduction to linear algebra (linear systems, Basis and dimension, linear maps, determinants).

The second semester includes the following topics:

1. Polynomial algebra (roots, divisibility and factorization, polynomials of several variables),

2. Vector spaces (direct sum, quotient spaces, linear forms and duality, bilinear and quadratic forms, Euclidean spaces, Hermitian forms and unitary spaces),

3. Linear transformations (eigenvectors, normal operators, Jordan canonical forms),

4. Affine and projective spaces (affine spaces, transformation and motions, Quadrics, projective space).