Google Search in:

## 2250 8:05am Lectures Week 6 S2015

Last Modified: February 25, 2015, 06:21 MST.    Today: December 14, 2018, 00:53 MST.
``` 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, 18 Dec 2013)Edwards-Penney 4.5, 4.6, 4.7 (7.0 K, txt, 18 Dec 2013)```
This is a 4-day week with Monday holiday.

### Wednesday: Cofactor Expansion, Inverse Formula, Determinant Product Theorem 3.6

``` Discussion of 3.5 problems.
Lecture
Ideas of rank, nullity, dimension in examples.Slides: Rank, nullity and elimination (156.3 K, pdf, 20 Dec 2012)
Lecture on Cofactor Method
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.
Adjugate Matrix.
How to form the matrix of cofactors and its transpose.
DEF: The adjugate matrix.
THEOREM. The 4 rules for computing any determinant can be
compressed into two rules,
det(triangular matrix)=the product of the diagonal elements,
det(EA)=det(E)det(A), where E is an elementary matrix, combo,
swap or mult.
Determinant product theorem
det(AB)=det(A)det(B) for any two square matrices A,B
Proof details.
Example.
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 2010 (227.1 K, pdf, 03 Mar 2012)Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)html: Problem notes S2015 (2.8 K, html, 01 Jan 2015)
REVIEW: Transpose matrix
(A^T)^T = A
(A + B)^T = A^T + B^T
(AB)^T = B^T A^T
det(A)=det(A^T)
Further properties of the adjugate matrix
Computing det(A) from A and adj(A) in 10 seconds
Problems involving adj(A): examples from exams.
Adjugate identity A adj(A) = adj(A) A = det(A) I
3x3 case: 6 ways to compute det(A) from A, adj(A).
3x3 case: the 6 cofactor expansions
```
```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, 20 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, 20 Dec 2012)Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)

```

### Wed-Fri: Vector spaces R^3 and R^n. Sections 4.1, 4.2.

```Exercises
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
Maple Answer Checks: Compute det(A), inverse(A), adjoint(A)
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

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
Triad i,j,k algebraic calculus model
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.
Head minus tail rule.
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
Intensity adjustments and scalar multiply
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)```

### Wed-Fri: 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 (168.4 K, pdf, 17 Feb 2014)
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 (168.4 K, pdf, 17 Feb 2014)Slides: Orthogonal vector theorem (124.8 K, pdf, 04 Dec 2014)
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 (168.4 K, pdf, 17 Feb 2014)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, 20 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, 04 Dec 2014)Transparencies: Ch4 Page 237+ slides, Exercises 4.1 to 4.4, some 4.9, from EP 2/E (463.2 K, pdf, 25 Sep 2003)html: Problem notes (2.8 K, html, 01 Jan 2015)```

### Thursday: Ziwen Zhu

``` Lab 5 due. Lab 6 introduced.
Sample Exam: Exam 1 and Exam 2 keys from S2014. See also S2012 and 2013, exams 1 and 2. The
first exam covers chapters 1,2,3.Sample Midterm 1, S2015, with solutions (1456.8 K, pdf, 19 Feb 2015)Exams and exam keys for the last 5 years (22.2 K, html, 23 Feb 2015)```

### 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.
Additional tests
Sampling test.
Wronskian test.
Orthogonal vector test.
THEOREM: Pivot columns are independent and non-pivot columns
are linear combinations of the pivot columns.
References for chapters 3 and 4, Linear Algebra
Manuscript: Linear algebraic equations, no matrices (429.7 K, pdf, 30 Jan 2014)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)Slides: Elementary matrix theorems (154.2 K, pdf, 11 Feb 2015)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, 14 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, 13 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 (97.0 K, pdf, 11 Feb 2015)Slides: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)Slides: Rank, nullity and elimination (156.3 K, pdf, 20 Dec 2012)Slides: Base atom, atom, basis for linear DE (117.6 K, pdf, 03 Mar 2012)Slides: Orthogonality (124.8 K, pdf, 04 Dec 2014)Slides: Partial fraction theory (148.6 K, pdf, 14 Dec 2014)Slides: The pivot theorem and applications (189.2 K, pdf, 03 Mar 2012)html: Problem notes S2015 (2.8 K, html, 01 Jan 2015)Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)Text: History of telecom companies (1.9 K, txt, 03 Apr 2013)

```