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.


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.


  • 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
a11 a12 a13
a21 a22 a23
a31 a32 a33

SVG file

Expansion by minors


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.