# 2250-4 12:55pm Lecture Record Week 5 S2010

Last Modified: February 14, 2010, 10:28 MST.    Today: September 24, 2018, 01:27 MDT.

## 9 Feb: Augmented Matrix for System Ax=b. RREF. Last Frame Algorithm. Sections 3.3, 3.4.

```Review
The three possibilities
Frame sequence analysis and the general solution.
Frame sequences with symbol k.
Last frame test.
Last frame algorithm.
Scalar form of the solution.
Preview: Vector form of the solution.
```
```Lecture: 3.3 and 3.4
Translation of equation models
Equality of vectors
Scalar equations translate to augmented matrix
Augmented matrix translate to scalar equations
Matrix toolkit: Combo, swap and multiply
Frame sequences for matrix models.
Special matrices
Zero matrix
identity matrix
diagonal matrix
upper and lower triangular matrices
square matrix
THEOREM. Homogeneous system with a unique solution.
THEOREM. Homogeneous system with more variables than equations.
Equation ideas can be used on a matrix A.
View matrix A as the set of coefficients of a homogeneous
linear system Ax=0. The augmented matrix B for this homogeneous
system would be the given matrix with a column of zeros appended:
B=aug(A,0).
matlab, maple and mathematica.
Pitfalls.
Answer checks should also use the online FAQ.
html: Problem notes S2010 (4.4 K, html, 31 Jan 2010)```

## Last Frame Algorithm

How to use maple to compute a frame sequence. Example is Exercise 3.2-14 from Edwards-Penney.
Frame sequences with symbol k.
```Matrices
Vector.
Matrix multiply
The college algebra definition
Examples.
Matrix rules
Vector space rules.
Matrix multiply rules.
Examples: how to multiply matrices on paper.
```

Slides: Matrix add, scalar multiply and matrix multiply (122.5 K, pdf, 02 Oct 2009)
Manuscript: Vectors and Matrices (266.8 K, pdf, 09 Aug 2009)
Manuscript: Matrix Equations (162.6 K, pdf, 09 Aug 2009)
```  General structure of linear systems.
Superposition.
General solution
X=X0+t1 X1 + t2 X2 + ... + tn Xn.
Matrix formulation Ax=b of a linear system
Properties of matrices: addition, scalar multiply.
Matrix multiply rules. Matrix multiply Ax for x a vector.
Linear systems as the matrix equation Ax=b.
```

## 9 Feb: Elementary matrices. Section 3.5

Lecture: Elementary matrices.
```  How to write a frame sequence as a matrix product
Fundamental theorem on frame sequences

THEOREM. If A1 and A2 are the first two frames of a sequence,
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 E1 to En such
that the frame sequence A ==> B can be written as
the matrix multiply equation
B=En En-1 ... E1 A.
```

## 11 Feb: Inverses. Rank and nullity. Section 3.5.

Discussion of 3.4 problems.
```How to compute the inverse matrix
Def: AB=BA=I means B is the inverse of A.
Frame sequences method.
Inverse rules
Web References: Construction of inverses. Theorems on inverses.

THEOREM. A square matrix A has a inverse if and only if
one of the following holds:
1. rref(A) = I
2. Ax=0 has unique solution x=0.
3. det(A) is not zero.
4. rank(A) = n =row dimension of A.
5. There are no free variables in the last frame.
6. All variables in the last frame are lead variables.
7. nullity(A)=0.

THEOREM. The inverse matrix is unique and written A^(-1).
THEOREM. If A, B are square and AB = I, then BA = I.
THEOREM. The inverse of inverse(A) is A itself.
THEOREM. If C and D have inverses, then so does CD and
inverse(CD) = inverse(D) inverse(C).
THEOREM.  The inverse of a 2x2 matrix is given by the formula

[a  b]        1    [ d  -b]
inverse [    ]  =  ------- [      ]
[c  d]     ad - bc [-c   a]

THEOREM.  The inverse B of any square matrix A can be
found from the frame sequence method

augment(A,I)
toolkit
steps combo, swap, mult
.
.
.
augment(I,B)

in which the inverse B of A is read-off from the right panel of
the last frame.

Slides: Inverse matrix, frame sequence method (71.6 K, pdf, 02 Oct 2009)Slides: Matrix add, scalar multiply and matrix multiply (122.5 K, pdf, 02 Oct 2009)```
```Elementary matrices
Inverses of elementary matrices.
Solving B=E3 E2 E1 A for matrix A = (E3 E2 E1)^(-1) B.
This problem is the basis for the fundamental theorem on
elementary matrices (see below). While 3.5-44 is a difficult
technical proof, the extra credit problems on this subject
replace the proofs by a calculation. See Xc3.5-44a and Xc3.5-44b.
How to do 3.5-16 in maple.
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); # Expected answer in right panel
evalm(A&*B);
```
```Lecture
Ideas of rank, nullity, dimension in examples.
Slides: Rank, nullity and elimination (111.6 K, pdf, 29 Sep 2009)   More on Rank, Nullity
dimension
3 possibilities
elimination algorithm
Question answered: What did I just do, by finding 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.
```

## 10-11 Feb: Murphy and Cox

Exam 1 review, questions and examples on exam problems 1,2,3,4,5.
Murphy's Lecture, time permitting: Maple Lab 2, problems 1,2,3 details.
The main lecture had a maple demo of the maple integration in problem 1, along with the introduction to the subject of the lab. This covered Newton Cooling, the hot tea example, and derivation of the DE used in maple lab 2.
Exam 1 date is Feb 17, 2-4pm in WEB 104 or Feb 18, 6:50am in JTB 140.
Sample Exam: Exam 1 key from F2009. See also S2009, exam 1.
Answer Key: Exam 1, f2009, 7:30am (62.9 K, pdf, 30 Sep 2009)
Answer Key: Exam 1, f2009, 12:25pm (339.3 K, pdf, 11 Oct 2009)
Answer Key: Exam 1, S2009, 7:30am (395.7 K, pdf, 02 Mar 2009)
Answer Key: Exam 1, S2009, 10:45am (310.2 K, pdf, 02 Mar 2009)

## 11 Feb: Determinants. Section 3.6.

See problem notes chapter 3
html: Problem notes F2009 (0.0 K, html, 31 Dec 1969)Lecture: Section 3.6, determinant theory and Cramer's rule.
Lecture: Methods for computing a determinant.
1. Sarrus' rule, 2x2 and 3x3 cases.
2. Four rules for determinants
1. Triangular Rule (one-arrow Sarrus' Rule): The determinant of a triangular matrix is the product of the diagonal elements.
2. Multiply rule: B=answer after mult(t,m), then |A| = (1/m) |B|
3. Swap rule: B=answer after swap(s,t), then |A| = (-1) |B|
4. Combo rule: B=answer after combo(s,t,c), then |A| = |B|
3. Cofactor expansion. Details for the 3x3 case.
4. Hybrid methods.
THEOREM. The 4 rules for computing any determinant can be compressed into two rules,
1. The triangular rule, and
2. det(EA)=det(E)det(A)
where E is an elementary combo, swap or mult matrix. Examples: Triangular rule [one-arrow Sarrus' rule], combo, swap and mult rules. Example: Cofactor rule.
```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).
```
<
```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 next time.
Hybrid methods to evaluate det(A) next time.
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
```
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 2008 (188.3 K, pdf, 26 Apr 2010)
Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)
html: Problem notes F2009 (0.0 K, html, 31 Dec 1969)