Exam 1 daySep 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.: Exam 1, F2009 and S2010 (17.7 K, html, 18 Dec 2010)Answer Keys

21 Sep: Augmented Matrix for System Ax=b. RREF. Last Frame Algorithm. Sections 3.3, 3.4.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 matrixTHEOREM. 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 F2010 (4.6 K, html, 26 Nov 2010)html## 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.

: Frame Sequence in maple, Exercise 3.2-14 (3.1 K, mws, 21 Aug 2010)Maple Worksheet: Frame Sequence in maple, Exercise 3.2-14 (2.8 K, txt, 23 Sep 2009)Maple Text: 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: Example 10 in Linear algebraic equations no matrices (292.8 K, pdf, 01 Feb 2010)ManuscriptMatricesVector. Matrix multiply The college algebra definition Examples. Matrix rules Vector space rules. Matrix multiply rules. Examples: how to multiply matrices on paper.: Matrix add, scalar multiply and matrix multiply (122.5 K, pdf, 02 Oct 2009)Slides: Vectors and Matrices (266.8 K, pdf, 09 Aug 2009)Manuscript: Matrix Equations (162.6 K, pdf, 09 Aug 2009)ManuscriptGeneral 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.

21 Sep: Elementary matrices. Section 3.5Elementary 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).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.

Web References: Elementary matrices: 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)Slides

22 Sep: MichalExam 1 daySep 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.: Exam 1, F2009 and S2010 (17.7 K, html, 18 Dec 2010)Answer Keys

23 Sep: Elementary Matrices. Inverses. Rank and nullity. Sections 3.4, 3.5.Discussion of3.4 problems.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. 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 algorithmSlidesQuestion 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 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)SlidesHow 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 3: Problem notes F2010 (4.6 K, html, 26 Nov 2010)html

28 Sep: Determinants. Sections 3.6.

slides for 3.6 determinant theory and Cramer's Rule: Determinants 2008 (167.7 K, pdf, 23 Sep 2010)Slides: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)ManuscriptMethods for computing a determinantSarrus' rule, 2x2 and 3x3 cases. Four rules for determinantsTriangular 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 (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