2250 7:30am Lectures Week 5 S2014

Last Modified: February 09, 2014, 21:32 MST.    Today: July 18, 2018, 12:34 MDT.
```Topics
Sections 3.4, 3.5, 3.6
The textbook topics, definitions, examples and theoremsEdwards-Penney 3.1, 3.2, 3.3 (3.0 K, txt, 19 Dec 2013)Edwards-Penney 3.4, 3.5, 3.6 (4.0 K, txt, 19 Dec 2013)```

Monday: Special matrices. Elementary matrices. Sections 3.3, 3.4, 3.5.

```Review of MatricesSlides: Matrix add, scalar multiply and matrix multiply (156.7 K, pdf, 03 Mar 2012)
Vector and vector operations.
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.
Theorem 1. Matrix algebra
A+B=B+A
A+(B+C)=(A+B)+C
A(BC)=(AB)C
A(B+C)=AB+AC, (D+E)F=DF+EF
Special matrices
diagonal matrix
upper and lower triangular matrices
square matrix
Def: Zero matrix 0
Theorems: 0+A=A+0, A0=0, 0A=0
Def: Identity matrix I
Theorems: AB=AC ==> B=C is false
AI=A, IB=B
Def: Augmented matrix of vectors A=[v1 v2 v3 ... vn] or  A=aug(v1 v2 v3 ... vn)
Same as Maple A:=< v1|v2|v3 > for n=3
Theorems: Ax in terms of columns of A
AB in terms of columns of B
Def: Matrix A has inverse matrix B means AB=BA=I
Inverse matrix
Definition: A has an inverse B if and only if AB=BA=I.
Theorem 1. An inverse is unique.
THEOREM 1a. If A has an inverse, then A is square.
Non-square matrices don't have an inverse.
THEOREM 1b. The zero matrix does not have an inverse.
Theorem 1. The inverse of a matrix is unique.
When it exists, then write B = A^(-1)
Theorem 2. Inverse of a 2x2 matrix A:=Matrix([[a,b],[c,d]])
equals B=(1/det(A))Matrix([[d,-b],[-c,a]])
Theorem 3. (a) (A^(-1))^(-1)=A
(b) (A^n)^(-1)=(A^(-1))^n for integer n>=0
(c) (AB)^(-1)=B^(-1) A^(-1)
Theorem 4. The nxn system Ax=b with A invertible has unique solution
x=A^(-1)b
Slides: Inverse matrix, frame sequence method (97.9 K, pdf, 03 Mar 2012)
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 to be answered: What did I just do, by finding rref(A)?
You solved the system Ax=0 by finding the Last Frame in a combo,
swap, mult sequence starting with matrix A or augmented matrix
aug(A,0). The RREF is the Last Frame. From it, use the Last
Frame Algorithm to find the general solution in scalar form,
then in vector form.
Problems 3.4-17 to 3.4-22 are homogeneous systems Ax=0 with matrix A
in reduced echelon form. Apply the last frame algorithm then write
the general solution in vector form.

EXAMPLE. A 3x3 matrix. Special Solutions of Gilbert Strang.
How to find the vector general solution.
How to find x_p: Set all free variable symbols to zero
How to find x_h: Take all linear combinations of the
Special Solutions.
Superposition: x = x_p + x_h = General Solution

Def: Elementary Matrix. It is constructed from the identity matrix I
by applying exactly one operation combo, swap or mult. Conventions
E = combo(I,s,t,c) [E is an elementary combo matrix]
E = swap(I,s,t)    [E is an elementary swap matrix]
E = mult(I,t,m)    [E is an elementary multiply matrix]
EXAMPLE: For the 2x2 identity I=Matrix([[1,0],[0,1]]),
E=mult(I,2,m)=Matrix([[1,0],[0,m]])

Definitions and details: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 sequence. Symbols
En, En-1, ... E1
are elementary square matrices that represent the operations combo,
swap, mult that created the sequence.
```

Tuesday: Inverses. Elementary matrices. Sections 3.4, 3.5.

```How to compute the inverse matrix
Def: AB=BA=I means B is the inverse of A.
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 sequence of frames
augment(A,I)
then toolkit operations
combo, swap, mult
to arrive at the Last Frame
augment(I,B)
The inverse of A equals matrix B in the right panel (last frame).
Slides: Inverse matrix, frame sequence method (97.9 K, pdf, 03 Mar 2012)Slides: Matrix add, scalar multiply and matrix multiply (156.7 K, pdf, 03 Mar 2012)Slides: Elementary matrix theorems (161.5 K, pdf, 03 Mar 2012)
Elementary matrices.
How to write a combo-swap-mult sequence as a matrix product
Fundamental theorems on combo-swap-mult sequences
THEOREM.
If B immediately follows A in a combo-swap-mult sequence,
then B = E A, where E is an elementary matrix having
EXACTLY ONE of the following forms:
E=combo(I,s,t,c), or
E= swap(I,s,t), or
E= mult(I,t,m)
Proof: See problem 3.5-39.
THEOREM.
If a combo-swap-mult sequence starts with matrix A and ends with
matrix 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.

THEOREM. Every elementary matrix E has an inverse. It is found
as follows:
Elementary Matrix   Inverse Matrix
E=combo(I,s,t,c)    E^(-1)=combo(I,s,t,-c)
E=swap(I,s,t)       E^(-1)=swap(I,s,t)
E=mult(I,t,m)       E^(-1)=mult(I,t,1/m)

Web References: Elementary matricesSlides: Elementary matrix theorems (161.5 K, pdf, 03 Mar 2012)
Inverses of elementary matrices.
PROBLEM. Solve B=E3 E2 E1 A for matrix A.
ANSWER. 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.
```

Wednesday: Determinants. Survey of Results. Section 3.6.

```  The textbook topics, definitions and theoremsEdwards-Penney 3.1, 3.2, 3.3 (3.0 K, txt, 19 Dec 2013)Edwards-Penney 3.4, 3.5, 3.6 (4.0 K, txt, 19 Dec 2013)
College Algebra Background:
College algebra determinant definition
Sarrus' rule for 2x2 and 3x3 matrices.
References for 3.6 determinant theory and Cramer's RuleSlides: Determinants 2008 (227.1 K, pdf, 03 Mar 2012)Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)
How to do 3.5-16 in maple.
A:=Matrix([[1,-3,-3],[-1,1,2],[2,-3,-3]]);
id:=<1,0,0|0,1,0|0,0,1>;
A1:= < A | id >;
linalg[rref](A1);    # inverse in right panel
A2:=combo(A1,1,2,1); # Doing the steps one at a time
A3:=combo(A2,1,3,-2);
WARNING. Some linalg functions return a matrix X with wrong format.
Be willing to use XX:=convert(X,Matrix), as needed. Or,
retire linalg and use only the LinearAlgebra package.
See problem notes chapter 3html: Problem notes S2014 (4.9 K, html, 10 Dec 2013)
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=mult(A,t,m), then |A| = (1/m) |B|
Swap rule: B=swap(A,s,t), then |A| = (-1) |B|
Combo rule: B=combo(A,s,t,c), then |A| = |B|
```

Thursday: Rebecca

```Lab4 due. Lab session on Lab5.
Sample Exam: Exam 1 key from S2012. See also F2010, exam 1.Sample Midterm 1, S2014, with solutions (1625.5 K, pdf, 11 Mar 2013)Exams and exam keys for the last 5 years (22.4 K, html, 30 Apr 2014)```