Introduction

Linear Algebra



Weiqi Zhou

Core content of this course

solving systems of linear equations

Example

  • There are some rabbits and chicken in the barn. What will be their respective numbers if there are 94 legs and 35 heads in total?

  • Suppose that there are number of chickens and number of rabbits, then we get

Exercise

  • solve the following linear systems:







Crucial parts of this course

  • A linear system may
    (1) admits a unique solution
    (2) has infinitely many solutions
    (3) admits no solution
    How do we judge and (if possible) solve it systematically

  • Linear maps: modern perspectives and notations

Objective of this course

At the end of the semester
you should be able to answer such questions

Structures of this course

  • （1）Vectors, linear dependence and their relations to linear systems

  • （2）Representing linear maps：matrices and their computations

  • （3）Solutions to linear systems (structures and approaches)

  • （4）Eigenpairs and diagonalization

Dependences

  • (1)-->(2)+(3)-->(4)

  • 1-3 lectures for each part

  • 1 exercise session for each part

Brief history

  • Solving and systems：thousand years ago
    Ancient Babylon | Nine Chapters on the Mathematical Art

  • Determinants：17th-18th century，by Leibniz, Cramer, Euler

  • Early forms and concepts of matrices: 19th century
    by Gauss, Sylvester, Cayley

  • Extensive use and developments: after middle 20th century

Summary

Leads and overview of this course