# 2250-2 7:30am Lecture Record Week 5 F2010

Last Modified: September 25, 2010, 06:28 MDT.    Today: September 24, 2018, 01:23 MDT.

## 20 Sep: Augmented Matrix for System Ax=b. RREF. Last Frame Algorithm. Sections 3.2, 3.3, 3.4.

```Review
The three possibilities
Frame sequence analysis and the general solution.
Last frame test.
Last frame algorithm.
Scalar 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
Problem session on ch3 problems.
How to use maple to make frame sequences. No solution example 3.1-16.
PDF: Maple frame sequence, no solution example (25.6 K, pdf, 20 Sep 2010)Maple Text: Maple code, frame sequence with no solution (0.2 K, mpl, 20 Sep 2010)
Answer checks should also use the online FAQ.html: Problem notes F2010 (4.6 K, html, 26 Nov 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.

Maple Worksheet: Frame Sequence in maple, Exercise 3.2-14 (3.1 K, mws, 21 Aug 2010)
Maple Text: Frame Sequence in maple, Exercise 3.2-14 (2.8 K, txt, 23 Sep 2009)
Slides: Three possibilities, theorems on infinitely many solutions, equations with symbols (101.0 K, pdf, 28 Sep 2010)
Beamer slides: 3 possibilities with symbol k (60.0 K, pdf, 31 Jan 2010)
Slides: 3 possibilities with symbol k (72.8 K, pdf, 31 Jan 2010)
Manuscript: Example 10 in Linear algebraic equations no matrices (292.8 K, pdf, 01 Feb 2010)
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)

## 21 Sep: Elementary matrices. Section 3.5

```Matrices
Vector.
Matrix multiply
The college algebra definition
Examples.
Matrix rules
Vector space rules.
Matrix multiply rules.
Examples: how to multiply matrices on paper.
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.
```
```Preview: 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).
Proof: See problem 3.5-39.
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.
```

## 22 Sep: Structure of linear systems. Rank and nullity. Sections 3.3, 3.4, 3.5.

``` 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.
```
``` General structure of linear systems.
Superposition.
General solution
X=X0+t1 X1 + t2 X2 + ... + tn Xn.
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.
```
```Discussion of 3.5 problems.
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
```

## 23 Sep: Laura

```Exam 1 day
Sep 23 at 7:25am in WEB 103. Exam problems 1,2,3 only.
Sample Exam: Exam 1 key from S2010. See also F2009, exam 1.
Answer Keys: Exam 1, F2009 and S2010 (17.7 K, html, 18 Dec 2010)
24 Sep: Elementary matrices. Inverses. Sections 3.4, 3.5.

Elementary matrices
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).
Proof: See problem 3.5-39.
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.
Inverses of elementary matrices.
Solving B=E3 E2 E1 A for matrix A = (E3 E2 E1)^(-1) B.
This problem uses the fundamental theorem on
elementary matrices (see above). 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 compute the inverse matrix
Def: AB=BA=I means B is the inverse of A.
Inverse = adjugate/determinant (2x2 case). See the theorem below for 2x2.
Inverse from the fundamental theorem on frame sequences.
Frame sequences method. See the theorem below.
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
1         [ d  -b]
-------    [        ]
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)  How to do 3.5-16 in maple.
with(linalg):#3.5-16
A:=matrix([[1,-3,-3],[-1,1,2],[2,-3,-3]]);