March 1, 2018

# Exam 1 notes

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

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
 a11 a12 a13 a21 a22 a23 a31 a32 a33

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.