# 2250-4 12:55pm Lecture Record Week 7 S2010

Last Modified: February 28, 2010, 13:08 MST.    Today: July 15, 2018, 13:30 MDT.

## 23 Feb: Review of 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 section 4.2).
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.
```

## 23 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.
Wronskian test.
Orthogonal vector test.
Pivot theorem.
Geometric tests.
One or two vector independence.
Geometry of dependence in dimensions 1,2,3.
```

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

```Additional Independence Tests
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.
```
```ANNOUNCEMENT:
Problem session 4.3, 4.4, 4.7 on Wed in WEB 104 and Thu in JTB 140.
Solutions to 4.3-18,24, 4.4-6,24 and 4.7-10,22,26.
Please refer to the chapter 4 problem notes.
```

## 24-25 Feb: Murphy

Exam 2 review.
Problem sessions on ch4 problems. Web 104 Wednesday and JTB 140 Thursday this week
```  How to construct solutions to 4.3-18,24, 4.4-6,24 and 4.7-10,22,26.
Questions answered on 4.3, 4.4, 4.7 problems.
Survey of 4.3 problems.
Illustration: How to do abstract independence arguments using vector
packages, without looking inside the packages.
Applications of the rank test and determinant test.
How to use the pivot theorem to identify independent vectors from a list.
```

## 25 Feb: Independence, basis and dimension

```MAPLE LAB 2. [laptop projection]
Solution to L2.2.
Graphic in L2.3.
Interpretation of graphics in L2.4.
PROBLEMS.
with(linalg):A:=matrix([[2,4,-3],[2,-3,-1],[-5,0,-3]]);
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)?
ALGEBRAIC 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..
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
BASIS.
Definition of basis and span.
Examples: Find a basis from a general solution formula.
Bases and the pivot theorem.
DIMENSION.
THEOREM. Two bases for a vector space V must have the same number of vectors.

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]
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.
```