向量、几何与线性代数

学分：6

本课程将讲述在各数学领域发现的大量重要概念及其应用，主要是在几何学领域的广泛应用。本课程还将讲述复数。复数对数学概念的统一，起到了深远的作用。

向量具备量值和方向特性，是描述线条和平面的自然方式，也成为在力学领域构建物理系统模型的合适语言。矩阵不但可以方便地解答复杂线性方程组，还可以在向量和坐标系统之间进行转化。

这反过来促进了线性代数和向量空间理论的诞生。向量空间概念的抽象化建立了另一个重要的、在数学领域中普遍存在的统合理论。该理论广泛应用于抽象群理论、视频游戏和信号处理技术。本课程将在课堂上讨论欧几里得平面坐标系并详述标准圆锥曲线理论。本课程还将讲述数学归纳法的基本验证方法。

在课程结课时，学生应能够：

• 阐明数学归纳法原理并演示其在典型证明中的应用

• 解答复数，进行标准计算，并理解复数与多项式根数和三角恒等式之间的关系

• 进行向量计算，包括数量积和向量积；运用向量解释直线与平面。

• 进行矩阵计算，包括化归至梯形矩阵并计算逆矩阵和行列式；通过矩阵法解答线性方程组；理解可逆矩阵与有解方程组之间的关系

• 说明向量空间概念及相关概念，如子空间和基体；计算给定例题中的维度和基体

• 理解线性变换的概念、计算给定基中的线性变换基体

• 通过平面上的笛卡儿坐标系和极坐标系找到简单曲线的参量表达式

• 认识、区分并表达各种形式的圆锥曲线

Vector, Geometry and Linear Algebra 向量、几何与线性代数

Credits: 6

This module introduces a number of powerful ideas found in all area of mathematics and its applications that are broadly geometric in flavour. Complex numbers, which turn out to underpin a profound unification of many ideas in mathematics, are introduced.

Vectors, which have both magnitude and direction, provide a natural way to describe lines and planes and are the appropriate language with which to model physical systems in mechanics. Matrices provide both a convenient way to deal with large systems of linear equations and to transform vectors and coordinate systems.

This in turn leads to the theory of linear algebra and vector spaces. The abstraction of the notion of a vector space is another powerful unifying theory that is found across mathematics, with applications in abstract group theory, video games and signal processing. Coordinate systems for the Euclidean plane are discussed and the standard theory of conic sections is developed. The module also introduces the fundamental proof technique of Mathematical Induction.

By the end of the module, students should be able to:

• State the Principle of Mathematical Induction and demonstrate its use in typical proofs.

• Work with complex numbers, perform standard calculations, and appreciate the relationship to the roots of polynomials and trigonometric identities.

• Perform vector calculations, including scalar and vector products, and describe lines and planes in terms of vectors.

• Perform matrix calculations including reducing to echelon form and calculating inverses and determinants, use matrix methods to solve systems of linear equations. Understand the relationship between invertible matrices and systems of equations with solutions.

• State the definition of a vector space and associated definitions of, for example, subspaces and bases. Calculate dimension and bases for given examples.

• Understand the notion of a linear transformation and calculate the matrix of a linear transformation with respect to a given basis.

• Use Cartesian and polar coordinate systems in the plane and find parametric expressions for simple curves.

• Recognise, classify and express conic sections in various forms.