# 2250-1 7:30am Lecture Record Week 5 S2012

Last Modified: February 14, 2012, 05:52 MST.    Today: July 15, 2018, 13:31 MDT.

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

```Review
The three possibilities
The Toolkit: combo, swap, mult
Frame sequences
Last frame test. The RREF.
Last frame algorithm: next time
Scalar equations and the augmented matrix
From equations to matrices and back again
The unique solution case
Scalar form of the unique solution.
The no solution case
```

## 7 Feb: Frame Sequence to RREF. Last Frame Algorithm. Sections 3.2, 3.3, 3.4.

```Review
The three possibilities
The Toolkit: combo, swap, mult
Frame sequences
Last frame test. The RREF.
1. Each nonzero equation has a lead variable
2. Lead variables appear in variable list order
3. Zero equations are last
Last frame algorithm
1. Apply the last frame test.
Proceed only if the system passes the test.
to the free variables.
4. Back-substitute the free variables into step 2.
5. Present the general solution as a list of variable
names in variable list order, with only invented
symbols on the right.
One long 35-minute example for a frame sequence
The unique solution case
Scalar form of the unique solution.
Why only lead variables in the last frame?
Why no free variables?
The no solution case
Signal equation
The infinitely many solution case
How to apply the last frame algorithm
A long example with one invented symbol
Examples with three invented symbols
Manuscript: Linear algebraic equations, no matrices (460.2 K, pdf, 10 Feb 2012)Slides: Linear equations, reduced echelon, three rules (233.6 K, pdf, 03 Mar 2012)Slides: Infinitely many solutions case (131.1 K, pdf, 03 Mar 2012)Slides: No solution case (79.7 K, pdf, 03 Mar 2012)Slides: Unique solution case (117.8 K, pdf, 03 Mar 2012)
```

## 8 Feb: Matrices. Translation Equation <==> Matrix. Sections 3.3,3.4,3.5

```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.
```
```  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 (42.2 K, pdf, 08 Feb 2012)Maple Text: Maple code, frame sequence with no solution (0.5 K, mpl, 08 Feb 2012)
Answer checks should also use the online FAQ.html: Problem notes S2012 (5.2 K, html, 08 Apr 2012)```

#### 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 (10.8 K, mws, 10 Feb 2012)Maple Text: Frame Sequence in maple, Exercise 3.2-14 (3.0 K, txt, 10 Feb 2012)Slides: Three possibilities, theorems on infinitely many solutions, equations with symbols (129.0 K, pdf, 03 Mar 2012)Beamer slides: 3 possibilities with symbol k (60.0 K, pdf, 31 Jan 2010)Slides: 3 possibilities with symbol k (98.8 K, pdf, 03 Mar 2012)Manuscript: Example 10 in Linear algebraic equations no matrices (460.2 K, pdf, 10 Feb 2012)Slides: Matrix add, scalar multiply and matrix multiply (156.7 K, pdf, 03 Mar 2012)Manuscript: Vectors and Matrices (426.4 K, pdf, 07 Feb 2012)Manuscript: Matrix Equations (292.1 K, pdf, 07 Feb 2012)```
```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.
```

## 9 Feb: Patrick

```Exam 1 day in one week
Review for Exam problems 1,2,3 only.
Sample Exam: Exam 1 key from F2010. See also S2010, exam 1.
Answer Keys: Exam 1, F2010 and S2010 (18.9 K, html, 02 May 2012)

10 Feb: Special matrices. Elementary matrices.  Sections 3.3, 3.4, 3.5.

Special matrices
Zero matrix
identity matrix
diagonal matrix
upper and lower triangular matrices
square matrix

Definition. Elementary Matrix.
Slides: Elementary matrix, the theory (161.5 K, pdf, 03 Mar 2012)
The purpose of introducing elementary matrices is to replace combo,
swap, mult frame sequences by matrix multiply equations of the form

B=En En-1 ... E1 A.

Symbols A and B stand for any two frames in a frame sequence. Symbols
En, En-1, ... E1
are square matrices that represent the operations combo, swap, mult that
already appeared in the frame sequence.

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.

Web References: Elementary matrices
Slides: Elementary matrix theorems (161.5 K, pdf, 03 Mar 2012)Slides: Elementary matrices, vector spaces (35.8 K, pdf, 18 Feb 2007)
Inverse matrix
Definition: A has an inverse B if and only if AB=BA=I.
THEOREM. An inverse is unique.
THEOREM. If A has an inverse, then A is square.
Non-square matrices don't have an inverse.
THEOREM. The zero matrix does not have an inverse.
THEOREM. Every elementary matrix E has an inverse. It is found
as follows:
Elementary Matrix     Inverse Matrix
combo(s,t,c)          combo(s,t,-c)
mult(t,m)             mult(t,1/m)
swap(s,t)             swap(s,t)

Inverses of elementary matrices.
Solving B=E3 E2 E1 A for matrix A = (E3 E2 E1)^(-1) B.