# 2250-4 12:55pm Lecture Record Week 6 S2010

Last Modified: February 20, 2010, 14:10 MST.    Today: July 15, 2018, 13:31 MDT.

## 16 Feb: Determinants, Cramers Rule, Adjugate formula. 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
Adjugate formula: A adj(A) = adj(A) A = det(A) I
```
```REVIEW
Cofactor expansion of det(A).
minor(A,i,j)
checkerboard sign (-1)^{i+j}
cofactor(A,i,j)=(sign)minor(A,i,j)
Special theorems for det(A)=0
a zero row or col
duplicates rows
proportional rows.
Elementary matrices
Determinant product rule for an elementary matrix
```
```Cofactor Expansion
Expansion 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.
How to form the matrix of cofactors and its transpose.
```
```Frame sequences and determinants
Formula for det(A) in terms of swap and mult operations.
```
``` Cramer's rule
How to solve Ax=b:
x1 = delta1/delta, ... , xn = deltan/delta
```
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.
```
```Adjugate matrix theorems
THEOREM. The adjugate formula A adj(A) = adj(A) A = det(A) I.
THEOREM. Adjugate inverse formula: inverse(A) = adj(A)/det(A).
```

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)
```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.
```

## 116 Feb: Introduction to Chapter 4. Vector Space. Section 4.1.

```Four Vector Models:
Fixed vectors
Triad i,j,k algebraic calculus model
Physics and Engineering arrows
Gibbs vectors.
Slides: vector models and vector spaces (110.3 K, pdf, 03 Oct 2009)   Parallelogram law.
Head minus tail rule.
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
Intensity adjustments and scalar multiply
```
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)

## 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)

## References for chapters 3 and 4, Linear Algebra

Manuscript: Linear algebraic equations, no matrices (292.8 K, pdf, 01 Feb 2010)
Slides: vector models and vector spaces (110.3 K, pdf, 03 Oct 2009)
Manuscript: Linear equations, reduced echelon, three rules (45.8 K, pdf, 22 Sep 2006)
Manuscript: Three rules, frame sequence, maple syntax (35.8 K, pdf, 25 Jan 2007)
Manuscript: Vectors and Matrices (266.8 K, pdf, 09 Aug 2009)
Manuscript: Matrix Equations (162.6 K, pdf, 09 Aug 2009)
Transparencies: Ch3 Page 149+, Exercises 3.1 to 3.6 (869.6 K, pdf, 25 Sep 2003)
Transparency: Sample solution ER-1 [same as L3.1] (184.6 K, jpg, 08 Feb 2008)
Slides: Elementary matrix theorems (114.4 K, pdf, 03 Oct 2009)
Slides: Elementary matrices, vector spaces (35.8 K, pdf, 18 Feb 2007)
Slides: Three possibilities, theorems on infinitely many solutions, equations with symbols (100.3 K, pdf, 23 Sep 2009)
Beamer slides: 3 possibilities with symbol k (60.0 K, pdf, 31 Jan 2010)
Slides: 3 possibilities with symbol k (72.8 K, pdf, 31 Jan 2010)
Slides: Linear equations, reduced echelon, three rules (155.6 K, pdf, 06 Aug 2009)
Slides: Infinitely many solutions case (93.8 K, pdf, 03 Oct 2009)
Slides: No solution case (58.4 K, pdf, 03 Oct 2009)
Slides: Unique solution case (86.0 K, pdf, 03 Oct 2009)
Maple: Lab 5, Linear algebra (94.2 K, pdf, 01 Jan 2010)
Slides: Three rules, frame sequence, maple syntax (35.8 K, pdf, 25 Jan 2007)
Transparencies: 3x3 Frame sequence and general solution (90.0 K, pdf, 28 Sep 2006)
html: Problem notes S2010 (4.4 K, html, 31 Jan 2010)
Slides: Determinants 2008 (188.3 K, pdf, 26 Apr 2010)
Manuscript: Determinants, Cramers rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)
Slides: Matrix add, scalar multiply and matrix multiply (122.5 K, pdf, 02 Oct 2009)
Slides: Digital photos, Maxwell's RGB separations, visualization of matrix add (153.7 K, pdf, 16 Oct 2009)
Slides: Inverse matrix, frame sequence method (71.6 K, pdf, 02 Oct 2009)
Slides: More on digital photos, checkerboard analogy (109.5 K, pdf, 02 Oct 2009)
Slides: Rank, nullity and elimination (111.6 K, pdf, 29 Sep 2009)
Slides: Base atom, atom, basis for linear DE (85.4 K, pdf, 20 Oct 2009)
Slides: Orthogonality (87.2 K, pdf, 10 Mar 2008)
Slides: Partial fraction theory (121.5 K, pdf, 30 Aug 2009)
Slides: The pivot theorem and applications (131.9 K, pdf, 02 Oct 2009)
Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)
Text: History of telecom companies (1.4 K, txt, 30 Dec 2009)

## 18 Feb: Subspace Tests and Applications. Sections 4.2, 4.3.

```  Data recorder example.
A certain planar kinematics problem records the data set V  using
three components x,y,z. The working set S is a plane described by
an ideal equation ax+by+cz=0. This plane is the hidden subspace of
the physical application, obtained by a computation on the original
data set V.
More on vector spaces and subspaces:
Detection of subspaces and data sets that are not subspaces.
Theorems:
Subspace criterion,
Kernel theorem,
Not a subspace theorem.
Use of theorems 1,2 in section 4.2.
Problem types in 4.1, 4.2.
Example:
Subspace Shortcut for the set S in R^3 defined by x+y+z=0.
Avoid using the subspace criterion on S, by writing it as Ax=0,
followed by applying the kernel theorem (thm 2 page 239 or 243
section 4.2 of Edwards-Penney).
Subspace applications.
When to use the kernel theorem.
When to use the subspace criterion.
When to use the not a subspace theorem.
Problems 4.1,4.2.
```

## 18 Feb: Independence and Dependence. Sections 4.1, 4.4, 4.5, 4.7

Lecture: Sections 4.1, 4.4 and some part of 4.7.
```Drill:
The 8-property vector toolkit.
Example: Prove zero times a vector is the zero vector.
The kernel: Solutions of Ax=0.
Find the kernel of the 2x2 matrix with 1 in the upper
right corner and zeros elsewhere.
```
Review of Vector spaces.
```  Vectors as packages of data items. Vectors are not arrows.
Examples of vector packaging in applications.
Fixed vectors.
Gibbs motions.
Physics i,j,k vectors.
Arrows in engineering force diagrams.
Functions, solutions of DE.
Matrices, digital photos.
Sequences, coefficients of  Taylor and Fourier series.
Hybrid packages.
The toolkit of 8 properties.
Subspaces.
Data recorder example.
Data conversion to fit physical models.
Subspace criterion (Theorem 1, 4.2).
Kernel theorem (Theorem 2, 4.2).
Not a Subspace Theorem.
```
Lecture: Independence and dependence.
``` Example: c1 e^x+ c2 xe^{-x} = 2 e^x + 3 e^{-x} ==> c1=2, c2=3.
Solutions of differential equations are vectors.
Geometric tests
One vector v1.
Two vectors v1, v2.
Algebraic tests.
Rank test.
Determinant test.
Sampling test.