Day 2: Linear Algebra

• Last Time: Welcome

• Announcements:
  • Departmental picnic: Saturday, 9/17 (Tower Park, Ft. Thomas)
  • Assignment for next time: Read 1.2

  The problem assignments can be found on the assignments page, although I will not put "reading" assignments: as I said, it goes without saying that, if we are discussing section X on a given day, then I will have expected you to have read section X....

• Brief motivation:
  • Vectors - 2-d, 3-d, ..., n-d?!
  • Matrices
    • spreadsheets
    • images
    • linear transformations, e.g. averages, rotations, ...
    • decompositions (Eharmony, statistical techniques)

• Today:
  • Linear systems:
    • A little system:
      • • How did you do it back in high school?
        • How do we do it here?
          • Eliminate, then
          • back-substitute.

    • Section 1.1 summary
      • some systems are easy...
        • "diagonal"
      • some systems may seem "silly":
        • inconsistent
        • redundent
    • The geometry of linear systems
      • exactly one solution
      • no solution
      • infinitely many solutions

  • Matrices allow us to eliminate unnecessary stuff! They suppress variable names, which are irrelevant to the solution.

  • Elementary row operations:
    • The point: Replace a system by a system with the same solution set that's easier to solve
    • What's legal?
      • replacement of a row by a sum of the row and multiples of other rows
      • interchange of rows
      • scaling of a row by a nonzero constant

  • Existence and Uniqueness
    • Is there at least one solution? (consistent)
    • Is the solution unique?

  • Examples:
    • p. 11, #12, 15, 20, 26, 32

• Next time: Echelon forms

Links:

• Octaveweb