ReviewThe three possibilities Frame sequence analysis and the general solution. Last frame test. Last frame algorithm. Scalar form of the solution.

Lecture: 3.3 and 3.4Translation 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 Problem session on ch3 problems. How to use maple to make frame sequences. No solution example 3.1-16.: Maple frame sequence, no solution example (25.6 K, pdf, 20 Sep 2010): Maple code, frame sequence with no solution (0.2 K, mpl, 20 Sep 2010) Answer checks should also use the online FAQ.Maple Text: Problem notes F2010 (4.6 K, html, 26 Nov 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. 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.

Preview:Elementary matrices.How to write a frame sequence as a matrix product Fundamental theorem on frame sequencesTHEOREM. 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 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.

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.

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

Discussion of 3.5 problems. 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

Exam 1 daySep 23 at 7:25am in WEB 103. Exam problems 1,2,3 only.

Sample Exam: Exam 1 key from S2010. See also F2009, exam 1.: Exam 1, F2009 and S2010 (17.7 K, html, 18 Dec 2010)Answer Keys

24 Sep: Elementary matrices. Inverses. Sections 3.4, 3.5.Elementary matricesFundamental theorem on frame sequencesTHEOREM. 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 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.Inverses of elementary matrices.Solving B=E3 E2 E1 A for matrix A = (E3 E2 E1)^(-1) B.About problem 3.5-44This 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.How to compute the inverse matrixDef: AB=BA=I means B is the inverse of A. Inverse = adjugate/determinant (2x2 case). See the theorem below for 2x2. Inverse from the fundamental theorem on frame sequences. Frame sequences method. See the theorem below. 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 1 [ d -b] ------- [ ] 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); See problem notes chapter 3Slides: Problem notes F2010 (4.6 K, html, 26 Nov 2010)html## 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 (101.0 K, pdf, 28 Sep 2010)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 (170.1 K, pdf, 17 Aug 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 F2010 (4.6 K, html, 26 Nov 2010)html: Determinants 2008 (167.7 K, pdf, 23 Sep 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 (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)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