Week 5: 28 Sep, |
29 Sep, | 30 Sep, | 01 Oct, | 02 Oct, |

How to write a frame sequence as a product of elementary matrices.

Fundamental theorem on frame sequences

THEOREM. If A2 is the frame just after frame 1, A1, then A2=E A1 where E is the elementary matrix built from the identity matrix I by applying one toolkit operation combo(s,t,c), swap(s,t) or mult(t,m). THEOREM. If a frame sequence starts with A and ends with B, then B=(product of elementary matrices)A. The meaning: If A is the first frame and B a later frame in a sequence, then there are elementary swap, combo and mult matrices EWeb References: Elementary matrices_{1}to E_{n}such that the frame sequence A ==> B can be written as the matrix multiply equation B=E_{n}E_{n-1}... E_{1}A.

Due today, maple lab L2.1.

Lecture: How to compute the inverse matrix from inverse = adjugate/determinant (2x2 case) and also by frame sequences. Inverse rules.

Web Reference: Construction of inverses. Theorems on inverses.

Elementary matrices. Inverses of elementary matrices.

Solving B=E3 E2 E1 A for matrix A = (E3 E2 E1)^(-1) B.

How to do 3.5-16 in maple. Maple answer checks.

> with(linalg):#3.5-16 > A:=matrix([[1,-3,-3],[-1,1,2],[2,-3,-3]]); > A1:=augment(A,diag(1,1,1)); > rref(A1); > B:=inverse(A); > A2:=addrow(A1,1,2,1); > A3:=addrow(A2,1,3,-2); > evalm(A&*B);

Lecture: Ideas of rank, nullity, dimension in examples.

More on Rank, Nullity, dimension, 3 possibilities, elimination algorithm.

Answer to the question: What did I just do, when I found rref(A)?

Problems 3.4-17 to 3.4-22 are homogeneous systems Ax=0 with A in reduced echelon form. Apply the last frame algorithm then write the general solution in vector form.

Sarrus' rule for 2x2 and 3x3. General Sarrus' rule with n-factorial arrows.

Lecture: Adjugate formula for the inverse. Review of Sarrus' Rules.

- Sarrus' rule, 2x2 and 3x3 cases.
- Four rules for determinants
- Triangular Rule
- Multiply rule
- Swap rule
- Combo rule

- Cofactor expansion. Details for the 3x3 case.
- Hybrid methods.

- The triangular rule, and
- det(EA)=det(A)det(A)

Review: College algebra determinant definition and Sarrus' rule for 2x2 and 3x3 matrices.

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).

- 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:

x_{1}= delta_{1}/delta, ... , x_{n}= delta_{n}/delta - Adjugate formula: A adj(A) = adj(A) A = det(A) I
- Adjugate inverse formula inverse(A) = adjugate(A)/det(A).

- Cofactor expansion of det(A).
- How to form minors, checkerboard signs and cofactors.

- 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 determinants having a zero row, duplicates rows or proportional rows.
- Elementary matrices and determinants. Determinant product rule for elementary matrices.
- Cramer's rule.

How to form the matrix of cofactors and its transpose, the adjugate matrix. - How to reduce the Four rules [triangular, swap , combo, mult] to Two Rules using the determinant product theorem det(AB)=det(A)det(B).

Exercises 3.4, 3.5 details.

Problems: 3.4-34 and 3.4-40. How to solve them. Cayley-Hamilton Theorem. Superposition proof. Web notes on these problems. Discussion of the Cayley-Hamilton theorem [Exercise 3.4-29; see also Section 6.3] Problem 3.4-29 is used in Problem 3.4-30. How to solve problem 3.4-30.The Cayley-Hamilton Theorem is a famous result in linear algebra which is the basis for solving systems of differential equations.

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.

Determinant product theorem det(EC)=det(E)det(C) for elementary matrics E det(AB)=det(A)det(B) for any two square matrices A,B Proof details. Example.Textbook: Chapter 4, sections 4.1 and 4.2.

Web references for chapter 4.

Lecture: Abstract vector spaces.Def: Vector==package of data items. Vectors are not arrows. The four vector models Fixed Triad i,j,k algebraic calculus model Physics and Engineering arrows Gibbs motion The 8-Property 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 Intensity adjustments and scalar multiply

- Digital photos and matrix add, scalar multiply visualization.

Parallelogram law. Head minus tail rule.

The 8-property toolkit for vectors. Vector spaces. Reading: Section 4.1 in Edwards-Penney, especially the 8 properties.

The 7:30 class lectures fell short of the target, while the 12:25 class remained ahead of schedule. What appears here for this week is accurate for only the 12:25 class.