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.

How 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)Slides

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 below). 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 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);

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.

- 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|

- Cofactor expansion. Details for the 3x3 case.
- Hybrid methods.

- The triangular rule, and
- det(EA)=det(E)det(A)

<THEOREM. Determinant values for elementary matrices: det(E)=1 for combo(s,t,c), det(E)=m for mult(t,m), det(E)=-1 for swap(s,t).

LectureCofactor expansion of det(A). minor(A,i,j) checkerboard sign (-1)^{i+j} cofactor(A,i,j)=(sign)minor(A,i,j) Details for 3x3 and 4x4 next time. Hybrid methods to evaluate det(A) next time. How to use the 4 rules to compute det(A) for any size matrix. Computing determinants of sizes 3x3, 4x4, 5x5 and higher. Frame sequences and determinants. Formula for det(A) in terms of swap and mult operations. Special theorems for det(A)=0 a zero row or col duplicates rows proportional rows. Elementary matrices Determinant product rule for an elementary matrix

- det(triangular matrix)=the product of the diagonal elements, and
- det(EA)=det(E)det(A), where E is an elementary combo, swap or mult matrix.

Determinant product theoremdet(AB)=det(A)det(B) for any two square matrices A,B Proof details. Example.