Day 10 Outline

March 1, 2018

Exam 1 notes

Assigned seating. Your individualized paper has your UBIT username on the front page. They will be laid out alphabetically by UBIT username.

exam1_cover_sample.png

Will need to bring your ID.

Points awarded for "style".

2.9 Dimension and rank

rank A = dim Col A

Rank theorem: rank A + dim Nul A = n

Basis theorem: If dim H = p, any LI set of vectors in H is a basis for H. And any spanning set of p vectors in H is a basis for H.

Invertible matrix theorem, cont'd

3.1 & 3.2 Determinants

More observations about ad-bc for

a b
c d

to 3x3 and higher.

Motivation (3.3): observations about determinant in Strang house - meaning?

reflections, rotations, scrunch, etc.

Ideas:

  • should be zero if there's a zero row
  • flip sign if rows swapped
  • multiplication of a single row (or column)
  • no change if multiple of one row added to other
2x2_colors_shapes.png
a11 a12 a13
a21 a22 a23
a31 a32 a33
3x3_colors_shapes.png

SVG file

Expansion by minors

cofactors

Determinant of a triangular matrix

Determinant via row-reduction

Determinant of transpose

Determinant of product AB, proof via writing A as product of elementary matrices

Answer some of the exercises in the book.