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Linear Algebra and Multivariable Calculus (Math 290) — Fall 2018

Welcome to Math 290! This is a three-quarter sequence in linear algebra and (you guessed it) multivariable calculus. The goal of the course is to learn how to use mathematical tools to solve problems in multiple variables and to understand why they work. In particular, this is not a proof-based course—the proof-based equivalent is Math 291, and higher-level courses in mathematical analysis will also derive many of the results we use in this course.

In Math 290-1 (Fall quarter) we will focus on linear algebra. Algebraically speaking, a linear problem is one that can be expressed using polynomials of degree ≤ 1—that is, variables cannot be multiplied together or raised to a power, and they can only be scaled by constants. (For example, 3x + 2y = 4 is a linear equation, but xy + sin(x2) = ey is not.) The reason for the word ‘linear’ is that when you translate such a problem to a geometric problem, the equations and constraints define points, lines, planes, and higher-dimensional hyperplanes. This translation between algebra and geometry turns out to be very fruitful; for example, we can simplify systems of linear equations by transforming the lines and planes they describe into new lines and planes. Such transformations are called linear maps and can be described using grids of numbers, called matrices. This is where this course comes in—most of our time will be devoted to studying matrices, developing algebraic tools for manipulating matrix equations, and using matrices to mediate between algebra and geometry.

Most administrative aspects of the course will be handled using Canvas.

Time and place. MoWe(Th)Fr 9:00–9:50am in Technological Institute L170.

Syllabus. You can download the course syllabus here.

Textbook. Our textbook will be Linear Algebra with Applications (5th edition) by Otto Brescher.

Homework. Homework assignments will be due at the beginning of class on Fridays—they can be viewed on Canvas.

Examinations. There will be two midterm exams and a final exam.

Handouts. What follows are the handouts from class and solutions to exercises.

  1. Linear systems — handoutsolutions
  2. Matrices and Gauss–Jordan elimination — handoutsolutions
  3. Reduced row-echelon form and rank — handoutsolutions
  4. Vector and matrix algebra — handoutsolutions
  5. Linear transformations — handoutsolutions
  6. Linear transformations in geometry — handoutsolutions
  7. Matrix products — handoutsolutions
  8. Inverse matrices — handoutsolutions
  9. More inverse matrices — handoutsolutions
  10. Images and kernels — handoutsolutions
  11. Midterm 1 reviewhandoutsolutions
  12. Subspaces, linear independence and bases — handoutsolutions
  13. More linear independence and bases — handoutsolutions
  14. Dimension and the rank–nullity theorem — handoutsolutions
  15. Coordinate vectors — handoutsolutions
  16. Change of basis — handoutsolutions
  17. More change of basis — handoutsupplement
  18. Determinants — handoutsolutions
  19. More determinants — handoutsolutions
  20. Midterm 2 reviewhandoutsolutions
  21. Geometrical interpretation of determinants — handoutsolutions
  22. Eigenvectors and eigenvalues — handoutsolutions
  23. Geometric and algebraic multiplicity — handoutsolutions
  24. Diagonalisation — handoutsolutions
  25. Criteria for diagonalisability — handoutsolutions
  26. Complex numbers — handoutsolutions