# 2250-1 7:30am Lecture Record Week 7 S2012

Last Modified: February 25, 2012, 20:25 MST.    Today: September 24, 2018, 02:21 MDT.

### Week 7, Feb 20 to 24: Sections 4.1, 4.2, 4.3, 4.4, 4.5

This is a 4-day week with only three lectures and one exam day. Monday was a holiday, President's Day.

#### Tue, 21 Feb: Problem Session Ch 3. Introduction to Chapter 4. Vector Space. Section 4.1.

```REVIEW PROBLEMS Ch3
3.6-60: Reading on induction. Required details.
B_n = 2B_{n-1} - B_{n-2},  B_n = n+1
3.6-review: matrix A is 10x10 and has 92 ones. What's det(A)?
```
```       Problem 3.5-60a and 60b.
How to discover the relation B_n = 2 B_{n-1} - B_{n-2}
Induction proof in 3.5-60b.

Problems 3.3-10,20 using maple
Problems 3.4-20,30,34,40
Problems 3.5-16,26,44
Problems 3.6-6,20,32,40,60
Some problem details appear already in the online problem notes.
The lecture adds more details, and complete solutions in several cases.

Maple computation of det(A), inverse(A), adjoint(A)
```
```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 (326.0 K, pdf, 07 Feb 2012)  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. See FAQ for 3.4.
Explained there: How to solve problem 3.4-30.
For the 3.5-44 proof, see the 3.5 FAQ.
```
```Four Vector Models:
Fixed vectors
Physics and Engineering arrows
Gibbs vectors.
Slides: vector models and vector spaces (144.1 K, pdf, 03 Mar 2012)   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 (170.2 K, pdf, 03 Mar 2012)
Slides: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)

#### 22 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.
The Span 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.
```
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 (182.3 K, pdf, 03 Mar 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 (154.1 K, pdf, 03 Mar 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 S2012 (5.2 K, html, 08 Apr 2012)

#### 24 Feb: Independence and Dependence. Sections 4.1, 4.4, 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).
Span Theorem (Theorem 1, 4.3)
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.