ReviewThe 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 3.4Translation 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). Answer checks matlab, maple and mathematica. Pitfalls. Answer checks should also use the online FAQ.: Problem notes S2010 (4.4 K, html, 31 Jan 2010)html

Frame sequences with symbol k.

MatricesVector. Matrix multiply The college algebra definition Examples. Matrix rules Vector space rules. Matrix multiply rules. Examples: how to multiply matrices on paper.

General structure of linear systems. Superposition. General solution X=X_{0}+t_{1}X_{1}+ t_{2}X_{2}+ ... + t_{n}X_{n}. 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.

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 E_{1}to E_{n}such that the frame sequence A ==> B can be written as the matrix multiply equation B=E_{n}E_{n-1}... E_{1}A.

Elementary matricesInverses of elementary matrices. Solving B=E3 E2 E1 A for matrix A = (E3 E2 E1)^(-1) B.About problem 3.5-44This 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.

Lecture Ideas of rank, nullity, dimension in examples.: Rank, nullity and elimination (111.6 K, pdf, 29 Sep 2009) More on Rank, Nullity dimension 3 possibilities elimination algorithmSlides

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.

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.

Sample Exam: Exam 1 key from F2009. See also S2009, exam 1.

See problem notes chapter 3How to compute the inverse matrixDef: 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.: 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 A2:=addrow(A1,1,2,1); A3:=addrow(A2,1,3,-2); evalm(A&*B);Slides

- 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

: Linear algebraic equations, no matrices (292.8 K, pdf, 01 Feb 2010)**Manuscript**: vector models and vector spaces (110.3 K, pdf, 03 Oct 2009)**Slides**: 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)**Manuscript**: Ch3 Page 149+, Exercises 3.1 to 3.6 (869.6 K, pdf, 25 Sep 2003)**Transparencies**: Sample solution ER-1 [same as L3.1] (184.6 K, jpg, 08 Feb 2008)**Transparency**: 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 (100.3 K, pdf, 23 Sep 2009)**Slides**: 3 possibilities with symbol k (60.0 K, pdf, 31 Jan 2010)**Beamer 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)**Slides**: Lab 5, Linear algebra (94.2 K, pdf, 01 Jan 2010)**Maple**: Three rules, frame sequence, maple syntax (35.8 K, pdf, 25 Jan 2007)**Slides**: 3x3 Frame sequence and general solution (90.0 K, pdf, 28 Sep 2006)**Transparencies**: Problem notes S2010 (4.4 K, html, 31 Jan 2010)**html**: Determinants 2008 (188.3 K, pdf, 26 Apr 2010)**Slides**: Determinants, Cramers rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)**Manuscript**: 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 (87.2 K, pdf, 10 Mar 2008)**Slides**: Partial fraction theory (121.5 K, pdf, 30 Aug 2009)**Slides**: The pivot theorem and applications (131.9 K, pdf, 02 Oct 2009)**Slides**: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)**Text**: History of telecom companies (1.4 K, txt, 30 Dec 2009)**Text**