Linear Algebra & Geometry

Course Information

• Linear Algebra and Space Analytic Geometry (UoG11108.06)
• This is an English taught course for first-year undergraduate students.
• This course introduces the fundamental concepts, methods and theories of linear algebra, vector spaces and quadratic forms.
• Teaching QQ group (slides, lecture notes): 1042962612

Purpose of the Course:

This course aims to provide a foundation in linear algebra, vector spaces and quadratic forms to prepare students for their applications in engineering.

Requirements by the end of this course:

• perform linear operations, multiplication, transpose and invert a matrix;
• solve linear equations by Gaussian elimination;
• compute the determinant of a matrix, and use Cramer’s rule to solve linear equations;
• evaluate the rank of a matrix and state its significance;
• apply the equations of lines and planes in vector form, to establish the relationship of several lines and that between a line and a plane;
• explain the linear combination and the linear dependence of a vector set, including the rank and the concept of a maximum independent set of a vector set;
• establish a basis, dimension and coordinates for an n-dimensional vector space;
• interpret the solutions of system of linear homogeneous equations in the context of a vector space;
• explain the concepts and properties of eigenvalues and eigenvectors, and compute them;
• diagonalize a matrix; Inner product, and use Schmidt’s orthogonalization to construct an orthonormal basis;
• derive the canonical (standard) form of quadratic forms by invertible transformations and orthogonal transformations;
• explain the concept of linear space, including bases, dimension and coordinates;

Textbook:

• David C. Lay, Steven R. Lay, Judi J. McDonald, Linear Algebra and its Applications, 5th edition, Pearson, 2015.

Reference:

• David C. Lay, Steven R. Lay, Judi J. McDonald, 线性代数及其应用（原书第5版）, 机械工业出版设, 2018.
• Gilbert Strang, Linear Algebra and its Applications, 4th edition, Cengage, 2006.
• Erwin Krewsig, Advanced Engineering Mathematics, 10th edition, Wiley, 2011.
• Peter V O’Neil, Advanced Engineering Mathematics, 7th edition, Cengage, 2012.

Slides

• Chapter 0. Introduction Slide
• Chapter 1. Linear Equations in Linear Algebra
  • 1.1 System of Linear Equations Slide
  • 1.2 Row Reduction and Echelon Forms Slide
  • 1.3 Vector Equations Slide
  • 1.4 The Matrix Equation Ax = b Slide
  • 1.5 Solution Sets of Linear Systems Slide
• Chapter 2. Matrix Algebra
  • 2.1 Matrix Operations Slide
  • 2.2 The Inverse of a Matrix Slide
  • 2.3 Characterizations of Invertible Matrices Slide
  • 2.4 Partitioned Matrices Slide
• Chapter 3. Determinants
  • 3.1 Introduction to Determinants Slide
  • 3.2 Properties of Determinants Slide
  • 3.3 Cramer’s Rule, Volume, and Linear Transformations Slide
• Chapter 4. Vector Spaces
  • 4.1 Vector Spaces and Subspaces Slide
  • 4.2 Null Spaces, Column Spaces, and Linear Transformations Slide
  • 4.3 Linearly Independent Sets; Bases Slide
  • 4.4 Coordinate Systems Slide
  • 4.5 The Dimension of a Vector Space Slide
  • 4.6 Rank Slide
• Chapter 5. Eigenvalues and Eigenvectors
  • 5.1 Eigenvectors and Eigenvalues Slide
  • 5.2 The Characteristic Equation Slide
  • 5.3 Diagnoalization Slide
  • 5.4 Eigenvectors and Linear Transformations Slide
• Chapter 6. Orthogonality and Least Squares
  • 6.1 Inner Product, Length, and Orthogonality Slide
  • 6.2 Orthogonal Sets Slide
  • 6.3 Orthogonal Projections Slide
  • 6.4 The Gram-Schmidt Process Slide
• Chapter 7. Symmetric Matrices and Quadratic Forms
  • 7.1 Diagonalization of Symmetric Matrices Slide
  • 7.2 Quadratic Forms Slide