# 2250-4 12:25pm Lecture Record Week 5 F2010

Last Modified: September 24, 2010, 20:17 MDT.    Today: January 17, 2018, 19:11 MST.

## 20 Sep: Michal

```Exam 1 day
Sep 20 at 12:50pm in JWB 335. 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) 21 Sep: 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.
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
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 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)
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.
21 Sep: Elementary matrices. Section 3.5
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.
Web References: Elementary matrices
Slides: vector models and vector spaces (110.3 K, pdf, 03 Oct 2009)Slides: Elementary matrix theorems (114.4 K, pdf, 03 Oct 2009)Slides: Elementary matrices, vector spaces (35.8 K, pdf, 18 Feb 2007)

22 Sep: Michal
Exam 1 day
Sep 22 at 12:50pm in JWB 335. 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)

23 Sep: Elementary Matrices. Inverses. Rank and nullity.  Sections 3.4, 3.5.

Discussion of 3.4 problems.
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 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.
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.

23 Sep: Inverses. Sections 3.4, 3.5.

How to compute the inverse matrix
Def: AB=BA=I means B is the inverse of A.
Inverse = adjugate/determinant (2x2 case)
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)How to do 3.5-16 in maple.
with(linalg):#3.5-16
A:=matrix([[1,-3,-3],[-1,1,2],[2,-3,-3]]);
B:=inverse(A); # expected answer
A1:=augment(A,diag(1,1,1));
rref(A1); # Expected answer in right panel
evalm(A&*B);
See problem notes chapter 3
html: Problem notes F2010 (4.6 K, html, 26 Nov 2010)
28 Sep: Determinants. Sections 3.6.

slides 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|

References for chapters 3 and 4, Linear Algebra
Manuscript: Linear algebraic equations, no matrices (292.8 K, pdf, 01 Feb 2010)Slides: vector models and vector spaces (110.3 K, pdf, 03 Oct 2009)Manuscript: Linear equations, reduced echelon, three rules (45.8 K, pdf, 22 Sep 2006)Manuscript: Three rules, frame sequence, maple syntax (35.8 K, pdf, 25 Jan 2007)Manuscript: Vectors and Matrices (266.8 K, pdf, 09 Aug 2009)Manuscript: Matrix Equations (162.6 K, pdf, 09 Aug 2009)Transparencies:  Ch3 Page 149+, Exercises 3.1 to 3.6 (869.6 K, pdf, 25 Sep 2003)Transparency: Sample solution ER-1 [same as L3.1] (184.6 K, jpg, 08 Feb 2008)Slides: Elementary matrix theorems (114.4 K, pdf, 03 Oct 2009)Slides: Elementary matrices, vector spaces (35.8 K, pdf, 18 Feb 2007)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)Slides: Linear equations, reduced echelon, three rules (155.6 K, pdf, 06 Aug 2009)Slides: Infinitely many solutions case (93.8 K, pdf, 03 Oct 2009)Slides: No solution case (58.4 K, pdf, 03 Oct 2009)Slides: Unique solution case (86.0 K, pdf, 03 Oct 2009)Maple: Lab 5, Linear algebra (170.1 K, pdf, 17 Aug 2010)Slides: Three rules, frame sequence, maple syntax (35.8 K, pdf, 25 Jan 2007)Transparencies: 3x3 Frame sequence and general solution (90.0 K, pdf, 28 Sep 2006)html: Problem notes F2010 (4.6 K, html, 26 Nov 2010)Slides: Determinants 2008 (167.7 K, pdf, 23 Sep 2010)Manuscript: Determinants, Cramers rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)Slides: Matrix add, scalar multiply and matrix multiply (122.5 K, pdf, 02 Oct 2009)Slides: Digital photos, Maxwell's RGB separations, visualization of matrix add (153.7 K, pdf, 16 Oct 2009)Slides: Inverse matrix, frame sequence method (71.6 K, pdf, 02 Oct 2009)Slides: More on digital photos, checkerboard analogy (109.5 K, pdf, 02 Oct 2009)Slides: Rank, nullity and elimination (111.6 K, pdf, 29 Sep 2009)Slides: Base atom, atom, basis for linear DE (85.4 K, pdf, 20 Oct 2009)Slides: Orthogonality (78.9 K, pdf, 14 Oct 2010)Slides: Partial fraction theory (121.5 K, pdf, 30 Aug 2009)Slides: The pivot theorem and applications (132.5 K, pdf, 10 Oct 2010)Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)Text: History of telecom companies (1.4 K, txt, 30 Dec 2009)

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