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所在平台: Udemy |
课程主页: https://www.udemy.com/course/linear-algebra-and-geometry-1/
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课程名称：线性代数与几何 1 课程概述： 本课程涵盖了线性方程组、矩阵、向量及其几何意义，旨在帮助学生建立线性代数的基础知识。 第一章：线性方程组 - S1. 课程介绍 - S2. 一些基本概念：回顾高中的数学知识，并介绍新的概念。 - S3. 线性方程组及其几何直觉：学习线性方程及其几何背景。 - S4. 求解线性方程组：掌握高斯消元法及其变形，处理不同类型的线性方程组。 - S5. 在数学和自然科学中的应用：了解线性方程组在其他学科中的应用。 第二章：矩阵与行列式 - S6. 矩阵及其运算：学习矩阵的定义及运算规则。 - S7. 逆矩阵及代数性质：使用矩阵代数，了解逆矩阵的定义。 - S8. 初等矩阵及求逆方法：通过高斯-约旦消元法计算逆矩阵。 - S9. 线性系统与矩阵之间的关系。 - S10. 行列式的定义及其性质：应用行列式的运算法则和克拉默法则求解方程。 第三章：向量及其运算 - S11. 二维、三维及n维空间中的向量：学习向量的运算及图示。 - S12. R^n中的距离与范数：计算R^n中点之间的距离及向量的范数。 - S13. 点积、正交性及正交投影：定义点积并计算几何向量之间的角度。 - S14. 叉积、平行四边形及平行六面体的体积：理解三维行列式的几何意义。 第四章：解析几何中的直线与平面 - S15. R^2中的直线：学习不同的直线方程表示法及其转换。 - S16. R^3中的平面：多种平面方程的表示及其转换方法。 - S17. R^3中的直线：掌握直线方程的几种描述方法。 - S18. 线性系统的几何；直线与平面的交点。 - S19. 点、直线和平面之间的距离计算。 - S20. 关于下一课程的简介：了解后续课程的内容。 请确保与您的教授确认哪些课程内容需用于期末考试，因为要求可能因国家、大学及年份而异。详细的课程内容及所有视频和问题的文本已在资源文件中提供。
Linear Algebra and Geometry 1Systems of equations, matrices, vectors, and geometryChapter 1: Systems of linear equationsS1. Introduction to the courseS2. Some basic conceptsYou will learn: some basic concepts that will be used in this course. Most of them are known from high-school courses in mathematics, some of them are new; the latter will appear later in the course and will be treated more in depth then.S3. Systems of linear equations; building up your geometrical intuitionYou will learn: some basic concepts about linear equations and systems of linear equations; geometry behind systems of linear equations.S4. Solving systems of linear equations; Gaussian eliminationYou will learn: solve systems of linear equations using Gaussian elimination (and back-substitution) and Gauss-Jordan elimination in cases of systems with unique solutions, inconsistent systems, and systems with infinitely many solutions (parameter solutions).S5. Some applications in mathematics and natural sciencesYou will learn: how systems of linear equations are used in other branches of mathematics and in natural sciences.Chapter 2: Matrices and determinantsS6. Matrices and matrix operationsYou will learn: the definition of matrices and their arithmetic operations (matrix addition, matrix subtraction, scalar multiplication, matrix multiplication). Different kinds of matrices (square matrices, triangular matrices, diagonal matrices, zero matrices, identity matrix).S7. Inverses; Algebraic properties of matricesYou will learn: use matrix algebra; the definition of the inverse of a matrix.S8. Elementary matrices and a method for finding A inverseYou will learn: how to compute the inverse of a matrix with Gauss-Jordan elimination (Jacobi's method).S9. Linear systems and matricesYou will learn: about the link between systems of linear equations and matrix multiplication.S10. DeterminantsYou will learn: the definition of the determinant; apply the laws of determinant arithmetics, particularly the multiplicative property and the expansion along a row or a column; solving equations involving determinants; the explicite formula for solving of n-by-n systems of linear equations (Cramer's rule), the explicite formula for inverse to a non-singular matrix.Chapter 3: Vectors and their productsS11. Vectors in 2-space, 3-space, and n-spaceYou will learn: apply and graphically illustrate the arithmetic operations for vectors in the plane; apply the arithmetic operations for vectors in R^n.S12. Distance and norm in R^nYou will learn: compute the distance between points in R^n and norms of vectors in R^n, normalize vectors.S13. Dot product, orthogonality, and orthogonal projectionsYou will learn: definition of dot product and the way you can use it for computing angles between geometrical vectors.S14. Cross product, parallelograms and parallelepipedsYou will learn: definition of cross product and interpretation of 3-by-3 determinants as the volume of a parallelepiped in the 3-space.Chapter 4: Analytical geometry of lines and planesS15. Lines in R^2You will learn: several ways of describing lines in the plane (slope-intercept equation, intercept form, point-vector equation, parametric equation) and how to compute other kinds of equations given one of the equations named above.S16. Planes in R^3You will learn: several ways of describing planes in the 3-spaces (normal equation, intercept form, parametric equation) and how to compute other kinds of equations given one of the equations named above.S17. Lines in R^3You will learn: several ways of describing lines in the 3-space (point-vector equation, parametric equation, standard equation) and how to compute other kinds of equations given one of the equations named above.S18. Geometry of linear systems; incidence between lines and planesYou will learn: determine the equations for a line and a plane and how to use these for computing intersections by solving systems of equations.S19. Distance between points, lines, and planesYou will learn: determine the equations for a line and a plane and how to use these for computing distances.S20. Some words about the next courseYou will learn: about the content of the second course.Make sure that you check with your professor what parts of the course you will need for your final exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.A detailed description of the content of the course, with all the 222 videos and their titles, and with the texts of all the 175 problems solved during this course, is presented in the resource file "001 Outline_Linear_Algebra_and_Geometry_1.pdf" under video 1 ("Introduction to the course"). This content is also presented in video 1.
