# 2250-4 12:25pm Lecture Record Week 6 F2010

Last Modified: October 03, 2010, 10:07 MDT.    Today: July 17, 2018, 17:23 MDT.

## 27 Sep, 29 Sep, 30 Sep: Michal and Laura

Exam 1. Problems 4,5. Take the exam 27 Sep or 29 Sep in JWB 335 at 12:50pm, or 30 Sep in WEB 103 at 7:25am.
Sample Exam: Exam 1 keys from S2010 and F2009. See also S2009, exam 1.
HTML: Exam links for the past 5 years (17.7 K, html, 18 Dec 2010)

## 28 Sep: Determinants, Survey of Results. Section 3.6

```College Algebra Background:
College algebra determinant definition
Sarrus' rule for 2x2 and 3x3 matrices.
```
References for 3.6 determinant theory and Cramer's Rule
Slides: Determinants 2008 (167.7 K, pdf, 23 Sep 2010)
Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)
```Methods for computing a determinant
Sarrus' rule, 2x2 and 3x3 cases.
Four rules for determinants
Triangular Rule (one-arrow Sarrus' Rule): The determinant of
a triangular matrix is the product of the diagonal elements.
Multiply rule: B=answer after mult(t,m), then |A| = (1/m) |B|
Swap rule: B=answer after swap(s,t), then |A| = (-1) |B|
Combo rule: B=answer after combo(s,t,c), then |A| = |B|
Results on Determinants
Examples: Computing det(A) easily. When does det(A)=0?
THEOREM. Determinant values for elementary matrices:
det(E)=1 for combo(s,t,c),
det(E)=m for mult(t,m),
det(E)=-1 for swap(s,t).
```
```Survey of Main theorems:
Computation by the 4 rules, cofactor expansion, hybrid methods.
Determinant product theorem det(AB)=det(A)det(B).
Cramer's Rule for solving Ax=b:
x1 = delta1/delta, ... , xn = deltan/delta
```

## 28 Sep: Cramers Rule, Adjugate formula. Section 3.6

```Lecture
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.
```
THEOREM. The 4 rules for computing any determinant can be compressed into two rules,
1. det(triangular matrix)=the product of the diagonal elements, and
2. det(EA)=det(E)det(A), where E is an elementary combo, swap or mult matrix.
```Determinant product theorem
det(AB)=det(A)det(B) for any two square matrices A,B
Proof details.
Example.
```

Slides: Determinants 2010 (167.7 K, pdf, 23 Sep 2010)
Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)
html: Problem notes F2010 (4.6 K, html, 26 Nov 2010)

## 30 Sep: Problem Session Ch 3. Introduction to Chapter 4. Vector Space. Section 4.1.

```Adjugate Matrix
Transpose matrix and properties.
(A^T)^T = A
(A + B)^T = A^T + B^T
(AB)^T = B^TA^T
The adjugate matrix. How to form the matrix of cofactors and its transpose.
Proofs: Special theorems for det(A)=0
a zero row or col
duplicates rows
proportional rows
RREF(A) not the identity
Problem Details
Exercises 3.4-30. Problem 3.4-29 is used in Problem 3.4-30.
How to solve problem 3.4-30 without 3.4-29.
Cayley-Hamilton Theorem. Result of 3.4-28,29.
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 (186.5 K, pdf, 09 Aug 2009)  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.
Showed how to write up the proof of 3.4-40(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.
Exercises 3.4, 3.5, 3.6.
Solution to 3.5-44.
Introduction to Ch4
Vector space. Subspace.
Vectors
Def: Vector==package of data items.
Vectors are not arrows.
Examples of vectors:
Digital photos,
Fourier coefficients,
Taylor coefficients,
Solutions to DE. Example: y=2exp(-x^2) for DE y'=-2xy, y(0)=2.
```