# 2250-1 7:30am Lecture Record Week 6 S2010

Last Modified: February 20, 2010, 14:16 MST.    Today: August 18, 2018, 17:30 MDT.

## 16 Feb: Determinants, Survey of Results. Section 3.6

```Drill:
College algebra determinant definition
Sarrus' rule for 2x2 and 3x3 matrices.
The Four Rules
Triangular rule [one-arrow Sarrus' rule]
combo rule
swap rule
mult rule
Examples: Computing det(A) easily. When does det(A)=0?
```
```THEOREM. Determinant values for elementary matrices:
det(E)=1 for combo(s,t,c),
det(E)=m for mult(t,m),
det(E)=-1 for swap(s,t).
```
```Survey of Main theorems:
Computation by the 4 rules, cofactor expansion, hybrid methods.
Determinant product theorem det(AB)=det(A)det(B).
Cramer's Rule for solving Ax=b:
x1 = delta1/delta, ... , xn = deltan/delta
```

## 17 Feb: Cramers Rule, Adjugate formula. Section 3.6

```Lecture
Cofactor expansion of det(A).
minor(A,i,j)
checkerboard sign (-1)^{i+j}
cofactor(A,i,j)=(sign)minor(A,i,j)
Details for 3x3 and 4x4.
Hybrid methods to evaluate det(A).
How to use the 4 rules to compute det(A) for any size matrix.
Computing determinants of sizes 3x3, 4x4, 5x5 and higher.
Frame sequences and determinants.
Formula for det(A) in terms of swap and mult operations.
Special theorems for det(A)=0
a zero row or col
duplicates rows
proportional rows.
Elementary matrices
Determinant product rule for an elementary matrix
Cramer's rule.
How to form the matrix of cofactors and its transpose.
```
THEOREM. The 4 rules for computing any determinant can be compressed into two rules,
1. det(triangular matrix)=the product of the diagonal elements, and
2. det(EA)=det(E)det(A), where E is an elementary combo, swap or mult matrix.
```Determinant product theorem
det(AB)=det(A)det(B) for any two square matrices A,B
Proof details.
Example.
```

Slides: Determinants 2008 (188.3 K, pdf, 26 Apr 2010)
Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)
html: Problem notes S2010 (4.4 K, html, 31 Jan 2010)

## 17-18 Feb: Murphy

Exam 1. Take the exam Feb 17, 1:45pm-3pm in WEB 104 or Feb 18, 6:45am in JTB 140.
Sample Exam: Exam 1 key from F2009. See also S2009, exam 1.
Answer Key: Exam 1, f2009, 7:30am (62.9 K, pdf, 30 Sep 2009)
Answer Key: Exam 1, f2009, 12:25pm (339.3 K, pdf, 11 Oct 2009)
Answer Key: Exam 1, S2009, 7:30am (395.7 K, pdf, 02 Mar 2009)
Answer Key: Exam 1, S2009, 10:45am (310.2 K, pdf, 02 Mar 2009)

## 19 Feb: Problem Session Ch 3. Introduction to Chapter 4. Vector Space. Section 4.1.

```Problem Details
Exercises 3.4-34 and 3.4-40.
Cayley-Hamilton Theorem.
It is a famous result in linear algebra which is the basis for
solving systems of differential equations.
Discussion of the Cayley-Hamilton theorem [Exercise 3.4-29;
Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)  Superposition proof
Problem 3.4-40 is the superposition principle for the
matrix equation Ax=b. It is the analog of the differential
equation relation y=y_h + y_p.
Web notes on the problems.
Problem 3.4-29 is used in Problem 3.4-30.
How to solve problem 3.4-30.
Problem 3.5-60a and 60b.
How to discover the reation B_n = 2 B_{n-1} - B_{n-2}
Induction proof in 3.5-60b.
```
```Four Vector Models:
Fixed vectors
Physics and Engineering arrows
Gibbs vectors.
Slides: vector models and vector spaces (110.3 K, pdf, 03 Oct 2009)   Parallelogram law.
Vector Toolkit
The 8-property toolkit for vectors.
Vector spaces.
Reading: Section 4.1 in Edwards-Penney, especially the 8 properties.
Lecture: Abstract vector spaces.
Def: Vector==package of data items.
Vectors are not arrows.
The 8-Property Vector Toolkit
Def: vector space, subspace
Working set == subspace.
Data set == Vector space
Examples of vectors:
Digital photos,
Fourier coefficients,
Taylor coefficients,
Solutions to DE. Example: y=2exp(-x^2) for DE y'=-2xy, y(0)=2.
RGB color separation and matrix add
```
Digital photos and matrix add, scalar multiply visualization.
Slides: Digital photos, Maxwell's RGB separations, visualization of matrix add (153.7 K, pdf, 16 Oct 2009)
Slides: More on digital photos, checkerboard analogy (109.5 K, pdf, 02 Oct 2009)
Textbook: Chapter 4, sections 4.1 and 4.2.
Web references for chapter 4. Repeated below in ch3-ch4 references.
Slides: Vector space, subspace, independence (132.6 K, pdf, 03 Feb 2010)
Manuscript: Vector space, Independence, Basis, Dimension, Rank (268.7 K, pdf, 22 Mar 2010)
Slides: The pivot theorem and applications (131.9 K, pdf, 02 Oct 2009)
Slides: Rank, nullity and elimination (111.6 K, pdf, 29 Sep 2009)
Slides: Digital photos, Maxwell's RGB separations, visualization of matrix add (153.7 K, pdf, 16 Oct 2009)
Slides: More on digital photos, checkerboard analogy (109.5 K, pdf, 02 Oct 2009)
Slides: Orthogonality (87.2 K, pdf, 10 Mar 2008)
Transparencies: Ch4 Page 237+ slides, Exercises 4.1 to 4.4, some 4.9 (463.2 K, pdf, 25 Sep 2003)
html: Problem notes S2010 (4.4 K, html, 31 Jan 2010)