## 2250 7:30am Lectures Week 6 S2013

Last Modified: February 12, 2013, 05:45 MST.    Today: September 23, 2018, 13:49 MDT.
``` Edwards-Penney, sections 4.1, 4.2, 4.3, 4.4
The textbook topics, definitions and theoremsEdwards-Penney 4.1, 4.2, 4.3, 4.4 (5.6 K, txt, 24 Dec 2012)Edwards-Penney 4.5, 4.6, 4.7 (7.0 K, txt, 24 Dec 2012)```

### Monday: Problem Session 3.3, 3.4, 3.5, 3.6

```New Topics
Rank, Nullity, Dimension and Elimination for Equations
Three possibilities
Definitions: rank, nullity, dimension
Rank-Nullity theorem
Elimination algorithm
Examples
References:Slides: Rank, nullity and elimination (156.3 K, pdf, 21 Dec 2012)
Ch 3 PROBLEMS
Problems 3.3-10,20 using maple
Problem 3.4-20
Long details in FAQ 3.4
Problem 3.4-30, Cayley-Hamilton
Problem 3.4-29 is used in Problem 3.4-30.
See FAQ 3.4 for details
Cayley-Hamilton Theorem.
It is a famous result in linear algebra which is the basis
for solving systems of differential equations. DiscussionSlides: Cayley-Hamilton Theorem (100.5 K, pdf, 21 Dec 2012)Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)
Problem 3.4-40, Superposition proof
The problem is to prove the superposition principle for the
matrix equation Ax=b. It is the analog of the differential
equation relation y=y_h + y_p. Details in FAQ 3.4.
Problems 3.5-16,26,44
For the 3.5-44 proof, see the 3.5 FAQ.
Problems 3.6-6,20,32,40,60
3.6 FAQ for details and answer checks
Review 3.6. matrix A is 10x10 and has 92 ones. What's det(A)?
Problem 3.6-60, nxn determinants
(60a) B_n = 2B_{n-1} - B_{n-2}, by cofactor expansion
(60b) B_n = n+1 by induction
```

### Tuesday: Vector spaces R^3 and R^n. Sections 4.1, 4.2.

```Intro to Ch4
Def: Vector==package of data items
Vector Toolkit
The 8-property toolkit for vectors [4.2]
Reading: Sections 4.1, 4.2 in Edwards-Penney
Def: vector space, subspace
Data set == Vector space
Working set == subspace.
Examples of vectors:
Four classical vector models,
Vectors are not arrows
Fixed vectors
Physics and Engineering arrows
Gibbs vectors.
Digital photos,
Fourier coefficients,
Taylor coefficients,
Solutions to DE.
Example: y=2exp(-x^2) for DE y'=-2xy, y(0)=2.
Four Vector Models:Slides: vector models and vector spaces (144.1 K, pdf, 03 Mar 2012)
Parallelogram law.
Abstract vector spaces, 4.2.
Def: Vector==package of data items.
Vectors are not arrows
The 8-Property Vector Toolkit
Def: abstract vector space
Data set == Vector space
Def: Subspace of a vector space
Working set == smaller vector space = subspace
Vector space of color photographs
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 (170.2 K, pdf, 03 Mar 2012)Slides: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)```

### Wednesday: Subspace Tests and Applications. Sections 4.2, 4.3.

```  Subspaces
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.
Web referenceSlides: Vector space, subspace, independence (185.0 K, pdf, 21 Dec 2012)
More on vector spaces and subspaces
Detection of subspaces and data sets that are not subspaces.
Subspace Theorems:
Subspace criterion,
Kernel theorem,
Not a subspace theorem.
The Span Theorem.
Preview: Independence-Dependence Theorems
Determinant test
Rank test
Pivot theorem
Orthogonal vector theorem
Wronskian test for functions
Sample test for functions
Web referencesSlides: Vector space, subspace theorems, independence tests (185.0 K, pdf, 21 Dec 2012)Slides: Orthogonal vector theorem (124.8 K, pdf, 03 Mar 2012)
Use of subspace theorems 1,2 in section 4.2.
Subspace 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 (4.2 Theorem 2).
Subspace applications.
When to use the kernel theorem.
When to use the subspace criterion.
When to use the not a subspace theorem.
Identifying a subspace with the span theorem
Identifying a subspace defined by equations
Problems 4.1,4.2.

Textbook Reading: Chapter 4, sections 4.1 and 4.2.

Web references for chapter 4. Slides: Vector space, subspace, independence (185.0 K, pdf, 21 Dec 2012)Manuscript: Vector space, Independence, Basis, Dimension, Rank (268.7 K, pdf, 22 Mar 2010)Slides: The pivot theorem and applications (189.2 K, pdf, 03 Mar 2012)Slides: Rank, nullity and elimination (156.3 K, pdf, 21 Dec 2012)Slides: Digital photos, Maxwell's RGB separations, visualization of matrix add (170.2 K, pdf, 03 Mar 2012)Slides: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)Slides: Orthogonality (124.8 K, pdf, 03 Mar 2012)Transparencies: Ch4 Page 237+ slides, Exercises 4.1 to 4.4, some 4.9 (463.2 K, pdf, 25 Sep 2003)html: Problem notes S2013 (5.1 K, html, 06 Dec 2012)```

### Thursday: Leif

```Exam 1. Problems 1, 2, 3.
Sample Exam: Exam 1 keys from S2012 and F2010.HTML: Exam links for the past 5 years (19.6 K, html, 30 Apr 2013)```

### Friday: Independence and Dependence. Sections 4.1, 4.3, 4.7

```Sections 4.1, 4.3 and some part of 4.7.
Review:
Is the 8-property vector toolkit good for nothing?
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. This is a key example
in the theory of eigenanalysis.
```
Quick Review of Vector spaces 4.1, 4.2
```  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).
Span Theorem (Theorem 1, 4.3)
Not a Subspace Theorem (Theorem 1 backwards)
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.
Three vectors v1, v2, v3.
Abstract vector space tests
One vector v1.
Two vectors v1, v2.
Algebraic tests.
Rank test.
Determinant test.
Pivot theorem.
`Manuscript: Linear algebraic equations, no matrices (460.2 K, pdf, 10 Feb 2012)Slides: vector models and vector spaces (144.1 K, pdf, 03 Mar 2012)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 (426.4 K, pdf, 07 Feb 2012)Manuscript: Matrix Equations (292.1 K, pdf, 07 Feb 2012)Transparencies:  Ch3 all, Exercises 3.1 to 3.6 from EP 2nd Edition (869.6 K, pdf, 25 Sep 2003)Transparencies:  Ch4 all, Exercises 4.1 to 4.7, from EP 2nd edition (461.2 K, pdf, 03 Oct 2010)Transparency: Sample solution ER-1 [exam review and maple lab 3] (184.6 K, jpg, 08 Feb 2008)Slides: Elementary matrix theorems (161.5 K, pdf, 03 Mar 2012)Slides: Three possibilities, theorems on infinitely many solutions, equations with symbols (129.0 K, pdf, 03 Mar 2012)Beamer slides: 3 possibilities with symbol k (60.0 K, pdf, 31 Jan 2010)Slides: 3 possibilities with symbol k (98.8 K, pdf, 03 Mar 2012)Slides: Linear equations, reduced echelon, three rules (237.3 K, pdf, 15 Dec 2012)Slides: Infinitely many solutions case (131.1 K, pdf, 03 Mar 2012)Slides: No solution case (79.7 K, pdf, 03 Mar 2012)Slides: Unique solution case (110.1 K, pdf, 14 Dec 2012)Slides: Three rules, frame sequence, maple syntax (35.8 K, pdf, 25 Jan 2007)Jpeg: 3x3 Frame sequence and general solution (315.9 K, jpg, 12 Dec 2012)Slides: Determinants 2012 (227.1 K, pdf, 03 Mar 2012)Manuscript: Determinants, Cramers rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)Slides: Matrix add, scalar multiply and matrix multiply (156.7 K, pdf, 03 Mar 2012)Slides: Digital photos, Maxwell's RGB separations, visualization of matrix add (170.2 K, pdf, 03 Mar 2012)Slides: Inverse matrix, frame sequence method (97.9 K, pdf, 03 Mar 2012)Slides: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)Slides: Rank, nullity and elimination (156.3 K, pdf, 21 Dec 2012)Slides: Base atom, atom, basis for linear DE (117.6 K, pdf, 03 Mar 2012)Slides: Orthogonality (124.8 K, pdf, 03 Mar 2012)Slides: Partial fraction theory (160.7 K, pdf, 03 Mar 2012)Slides: The pivot theorem and applications (189.2 K, pdf, 03 Mar 2012)Maple: Lab 5, Linear algebra (170.0 K, pdf, 04 Dec 2012)html: Problem notes S2013 (5.1 K, html, 06 Dec 2012)Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)Text: History of telecom companies (1.9 K, txt, 04 Apr 2013)`