# 2250-1 7:30am Lecture Record Week 8 S2012

Last Modified: March 04, 2012, 03:02 MST.    Today: July 15, 2018, 13:28 MDT.

### Week 8, 27 Feb to 2 Mar: Sections 4.5, 4.6, 4.7, 5.1, 5.2, 5.3

#### Mon, Feb 27: Pivot Theorem. Independence Tests. Basis and Dimension. Sections 4.4, 4.5

```Additional Independence Tests
Sampling Test.
Wronskian test.
Orthogonal vector test.
Pivot theorem [this lecture].
THEOREM: Pivot columns are independent and non-pivot columns
are linear combinations of the pivot columns.
THEOREM: rank(A)=rank(A^T).
THEOREM: A set of nonzero pairwise orthogonal vectors is linearly independent.
Basis.
General solutions with a minimal number of terms.
Definition: Basis == independence + span.
Differential Equations: General solution and shortest answer.
Pivot Theorem.
Applications of the pivot theorem to find a largest set of independent vectors.
Maximum set of independent vectors from a list.
PROOFS. [slides]
The pivot theorem. Algorithm 2, section 4.5.
rank(A)=rank(A^T). Theorem 3, section 4.5.
DIGITAL PHOTOS.
Digital photos are matrices
Photos are vectors == data packages
Checkerboards and digital photos
Matrix add and RGB separation, visualization
Matrix scalar multiply, visualization
```

#### Tue, 28 Feb: Independence, basis and dimension

```
ALGEBRAIC INDEPENDENCE TESTS: mostly review
Rank test.
Determinant test.
Sampling test.
Wronskian test.
Orthogonal vector test.
Pivot theorem.
FUNCTIONS.
How to represent functions as graphs and as infinitely long column
vectors. Rules for add and scalar multiply. Independence tests
using functions as the vectors.
BASIS.
Definition of basis and span.
Examples: Find a basis from a general solution formula.
Bases and the pivot theorem.
DIMENSION.
THEOREM. Two independent sets, each of which span a subspace S of a
vector space V, must have the same number of vectors. This
unique number is called the DIMENSION of the subspace S.

Examples:
Last Frame Algorithm: Basis for a linear system Ax=0.
Last frame algorithm and the vector general solution.
Basis of solutions to a homogeneous system of linear algebraic equations.
Bases and partial derivatives of the general solution on the invented symbols t1, t2, ...
DE Example: y = c1 e^x + c2 e^{-x} is the general solution. What's the basis?
SOLUTION ATOMS and INDEPENDENCE.
Def. atom=x^n(base atom)
base atom = 1, exp(ax), cos(bx), sin(bx), exp(ax) cos(bx), exp(ax) sin(bx)
"atom" abbreviates "solution atom of a linear differential equation"
THEOREM. Atoms are independent.
EXAMPLE. Show 1, x^2, x^9 are independent
EXAMPLE. Show 1, x^2, x^(3/2) are independent [Wronskian test]
PROBLEM 4.7-26.
How to solve y''+10y'=0 for general solution y=c1 + c2 exp(-10x)
```

#### Web, Feb 29:Basis and dimension. Intro to Linear Differential Equations.

```PARTIAL FRACTION THEORY.
Examples.
top=x-1, bottom=(x+1)(x^2+1)
top=x-1, bottom=(x+1)^2(x^2+1)^2
Maple assist with convert(top/bottom,parfrac,x);
PROBLEM 4.7-26.
How to solve y''+10y'=0 for general solution y=c1 + c2 exp(-10x)
Outline of the general theory used to solve linear differential equations.
Order of a DE and the dimension of the solution space.
Euler's theorem.
Finding solution atoms for a basis.
PROOFS. [slides]
rank(A)=rank(A^T). Theorem 3, section 4.5.
How to prove pivot theorem from frame sequence facts.
ALGEBRAIC INDEPENDENCE TESTS: mostly review
Rank test.
Determinant test.
Sampling test.
Application to x^2,exp(x)
Wronskian test.
Application to 1,x,x^2,x^3
Orthogonal vector test.
Example: (1,1,0),  (1,-1,0), ((0,0,1)
Pivot theorem.
Example: Find the independent columns in a matrix.
Example: Find the maximum number of independent vectors in a list.
FUNCTIONS.
How to represent functions as graphs and as infinitely long column
vectors. Rules for add and scalar multiply. Independence tests
using functions as the vectors.
BASIS.
Definition of basis and span.
Examples: Find a basis from a general solution formula.
Bases and the pivot theorem.
Example: Find a basis for the row vectors in a matrix.
Example: Find a basis or the column vectors in a matrix.
Equivalence of bases.
Example: A subspace S contains vectors v1,v2 and also vectors w1,w2.
When are both v1,v2 and w1,w2 bases for S?
A computer test for equivalent bases.
DIMENSION.
THEOREM. Two independent sets, each of which span a subspace S of a
vector space V, must have the same number of vectors. This
unique number is called the DIMENSION of the subspace S.
Examples:
Last Frame Algorithm: Basis for a linear system Ax=0.
Last frame algorithm and the vector general solution.
Basis of solutions to a homogeneous system of linear algebraic equations.
Bases and partial derivatives of the general solution on the invented symbols t1, t2, ...
DE Example: y = c1 e^x + c2 e^{-x} is the general solution. What's the basis?
SOLUTION ATOMS and INDEPENDENCE.
Def. atom=x^n(base atom)
base atom = 1, exp(ax), cos(bx), sin(bx), exp(ax) cos(bx), exp(ax) sin(bx)
"atom" abbreviates "solution atom of a linear differential equation"
THEOREM. Atoms are independent.
EXAMPLE. Show 1, x^2, x^9 are independent
EXAMPLE. Show 1, x^2, x^(3/2) are independent [Wronskian test]
PROBLEM 4.7-26. Review of Chapter 1 methods.
How to solve y''+10y'=0 for general solution y=c1 + c2 exp(-10x)
```
Web References:
Slides: Orthogonality, CSB-inequality, Pythagorean identity (124.8 K, pdf, 03 Mar 2012)
Slides: The pivot theorem and applications (189.2 K, pdf, 03 Mar 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: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)
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)

#### Thu, March 1: Patrick

``` Exam 2 review and Problem session on ch4 problems.
Exam 2 review for problems 1,2,3.
How to construct solutions to 4.7-10,20,26.
Questions answered on Chapter 4 problems.
Survey of solution methods for 4.3, 4.4, 4.5 problems.
Illustration: How to do abstract independence arguments using vector
packages, without looking inside the packages.
Applications of the Sampling test and Wronskian test for functions.
How to use the pivot theorem to identify independent vectors from a list.
```

#### Fri, March 2: Introduction to higher order linear DE. Sections 5.1, 5.2.

```MAPLE LAB 2. [laptop projection]
Hand Solution to L2.2.
Graphic in L2.3.
Interpretation of graphics in L2.4.
Summary for Higher Order Differential Equations
Slides: Atoms, Euler's theorem, 7 examples (130.3 K, pdf, 03 Mar 2012)Slides: Base atom, atom, basis for linear DE (117.6 K, pdf, 03 Mar 2012)  EXAMPLE. The equation y'' +10y'=0. Review.
How to solve y'' + 10y' = 0 with chapter 1 methods. Midterm 1 problem 1(d).
Idea: Let v=x'(t) to get a first order DE in v and a quadrature equation x'(t)=v(t).
Solve the first order DE by the linear integrating factor method. Then insert
the answer into x'(t)=v(t) and continue to solve for x(t) by quadrature.
Vector space of functions: solution space of a differential equation.
A basis for the solution space of y'' + 10y'=0 is {1,exp(-10x)}
ATOMS.
Base atoms are 1, exp(a x), cos(b x), sin(b x), exp(ax)cos(bx), exp(ax)sin(bx).
DEFINITION: atom=x^n(base atom).
THEOREM. Atoms are independent.
THEOREM. Solutions of constant-coefficient homogeneous differential
equations are linear combinations of atoms.
PICARD'S THEOREM.  It says that nth order equations have a solution space of dimension n.
EULER'S THEOREM. It says y=exp(rx) is a solution of ay''+by'+cy=0 <==> r is
a root of the characteristic equation ar^2+br+c=0.
Shortcut: The characteristic equation can be synthetically formed from the
differential equation ay''+by'+cy=0 by the formal replacement
y ==> 1, y' ==> r, y'' ==> r^2.
EXAMPLE. The equation y''+10y'=0 has characteristic equation r^2+10r=0
with roots r=0, r=-10.
Then Euler's theorem says exp(0x) and exp(-10x) are solutions.
By vector space dimension theory, 1, exp(-10x) are a basis for
the solution space of the differential equation.
Then the general solution is
y = c1 (1) + c2 (exp(-10x)).

Survey of topics for this week.
Linear DE Slides.
Slides: Picard-Lindelof, linear nth order DE, superposition (181.5 K, pdf, 03 Mar 2012)  Slides: How to solve linear DE or any order (168.3 K, pdf, 03 Mar 2012)  Slides: Atoms, Euler's theorem, 7 examples (130.3 K, pdf, 03 Mar 2012)
Theory of Higher Order Constant Equations:
Homogeneous and non-homogeneous structure.
Superposition.
Picard's Theorem.
Solution space structure.
Dimension of the solution set.
Atoms.
Definition of atom.
Independence of atoms.
Euler's theorem.
Real roots
Non-real roots [complex roots].
How to deal with conjugate pairs of factors (r-a-ib), (r-a+ib).
The Euler formula exp(i theta)=cos(theta) + i sin(theta).
How to solve homogeneous equations:
Use Euler's theorem to find a list of n distinct solution atoms.
Examples:   y''=0, y''+3y'+2y=0, y''+y'=0, y'''+y'=0.

Second order equations.
Homogeneous equation.
Harmonic oscillator example y'' + y=0.
Picard-Lindelof theorem.
Dimension of the solution space.
Structure of solutions.
Non-homogeneous equation. Forcing term.
Nth order equations.
Solution space theorem for linear differential equations.
Superposition.
Independence and Wronskians. Independence of atoms.
Main theorem on constant-coefficient equations
THEOREM. Solutions are linear combinations of atoms.
Euler's substitution y=exp(rx).
Shortcut to finding the characteristic equation.
Euler's basic theorem:
y=exp(rx) is a solution <==> r is a root of the characteristic equation.
Euler's multiplicity theorem:
y=x^n exp(rx) is a solution <==> r is a root of multiplicity n+1 of the characteristic equation.
How to solve any constant-coefficient nth order homogeneous differential equation.
1. Find the n roots of the characteristic equation.
2. Apply Euler's theorems to find n distinct solution atoms.
2a. Find the base atom for each distinct real root. Multiply
each base atom by powers 1,x,x^2, ... until the number of
atoms created equals the root multiplicity.
2b. Find the pair of base atoms for each conjugate pair of
complex roots. Multiply each base atom by powers 1,x,x^2,
... until the number of atoms created equals the root
multiplicity.
3. Report the general solution as a linear combination of the n atoms.
Picard's Theorem for higher order DE and systems. Solution space
dimension.

```