ReviewThe three possibilities The Toolkit: combo, swap, mult Frame sequences Last frame test. The RREF. Last frame algorithm: next time Scalar equations and the augmented matrix From equations to matrices and back again The unique solution case Scalar form of the unique solution. The no solution case

ReviewThe three possibilities The Toolkit: combo, swap, mult Frame sequences Last frame test. The RREF. 1. Each nonzero equation has a lead variable 2. Lead variables appear in variable list order 3. Zero equations are last Last frame algorithm 1. Apply the last frame test. Proceed only if the system passes the test. 2. Isolate lead variables left. 3. Assign invented symbols (t1, t2, t3, ...) to the free variables. 4. Back-substitute the free variables into step 2. 5. Present the general solution as a list of variable names in variable list order, with only invented symbols on the right. One long 35-minute example for a frame sequence The unique solution case Scalar form of the unique solution. Why only lead variables in the last frame? Why no free variables? The no solution case Signal equation The infinitely many solution case How to apply the last frame algorithm A long example with one invented symbol Examples with three invented symbols Reading:

: Linear algebraic equations, no matrices (460.2 K, pdf, 10 Feb 2012)Manuscript: Linear equations, reduced echelon, three rules (233.6 K, pdf, 03 Mar 2012)Slides: Infinitely many solutions case (131.1 K, pdf, 03 Mar 2012)Slides: No solution case (79.7 K, pdf, 03 Mar 2012)Slides: Unique solution case (117.8 K, pdf, 03 Mar 2012)Slides

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.

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 (42.2 K, pdf, 08 Feb 2012): Maple code, frame sequence with no solution (0.5 K, mpl, 08 Feb 2012) Answer checks should also use the online FAQ.Maple Text: Problem notes S2012 (5.2 K, html, 08 Apr 2012)html

Frame sequences with symbol k: Frame Sequence in maple, Exercise 3.2-14 (10.8 K, mws, 10 Feb 2012)Maple Worksheet: Frame Sequence in maple, Exercise 3.2-14 (3.0 K, txt, 10 Feb 2012)Maple Text: Three possibilities, theorems on infinitely many solutions, equations with symbols (129.0 K, pdf, 03 Mar 2012)Slides: 3 possibilities with symbol k (60.0 K, pdf, 31 Jan 2010)Beamer slides: 3 possibilities with symbol k (98.8 K, pdf, 03 Mar 2012)Slides: Example 10 in Linear algebraic equations no matrices (460.2 K, pdf, 10 Feb 2012)Manuscript: Matrix add, scalar multiply and matrix multiply (156.7 K, pdf, 03 Mar 2012)Slides: Vectors and Matrices (426.4 K, pdf, 07 Feb 2012)Manuscript: Matrix Equations (292.1 K, pdf, 07 Feb 2012)Manuscript

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.

Exam 1 day in one weekReview for Exam problems 1,2,3 only.

Sample Exam: Exam 1 key from F2010. See also S2010, exam 1.: Exam 1, F2010 and S2010 (18.9 K, html, 02 May 2012)Answer Keys

10 Feb: Special matrices. Elementary matrices. Sections 3.3, 3.4, 3.5.Special matricesZero matrix identity matrix diagonal matrix upper and lower triangular matrices square matrixDefinition. Elementary Matrix.: 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=ESlides_{n}E_{n-1}... E_{1}A. Symbols A and B stand for any two frames in a frame sequence. Symbols E_{n}, E_{n-1}, ... E_{1}are square matrices that represent the operations combo, swap, mult that already appeared in the frame sequence.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.

Web References: Elementary matrices: Elementary matrix theorems (161.5 K, pdf, 03 Mar 2012)Slides: Elementary matrices, vector spaces (35.8 K, pdf, 18 Feb 2007)SlidesInverse matrix Definition: A has an inverse B if and only if AB=BA=I. THEOREM. An inverse is unique. THEOREM. If A has an inverse, then A is square. Non-square matrices don't have an inverse. THEOREM. The zero matrix does not have an inverse. THEOREM. Every elementary matrix E has an inverse. It is found as follows: Elementary Matrix Inverse Matrix combo(s,t,c) combo(s,t,-c) mult(t,m) mult(t,1/m) swap(s,t) swap(s,t)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.## References for chapters 3 and 4, Linear Algebra

: Linear algebraic equations, no matrices (460.2 K, pdf, 10 Feb 2012)Manuscript: vector models and vector spaces (144.1 K, pdf, 03 Mar 2012)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 (426.4 K, pdf, 07 Feb 2012)Manuscript: Matrix Equations (292.1 K, pdf, 07 Feb 2012)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 (161.5 K, pdf, 03 Mar 2012)Slides: Elementary matrices, vector spaces (35.8 K, pdf, 18 Feb 2007)Slides: Three possibilities, theorems on infinitely many solutions, equations with symbols (129.0 K, pdf, 03 Mar 2012)Slides: 3 possibilities with symbol k (60.0 K, pdf, 31 Jan 2010)Beamer slides: 3 possibilities with symbol k (98.8 K, pdf, 03 Mar 2012)Slides: Linear equations, reduced echelon, three rules (233.6 K, pdf, 03 Mar 2012)Slides: Infinitely many solutions case (131.1 K, pdf, 03 Mar 2012)Slides: No solution case (79.7 K, pdf, 03 Mar 2012)Slides: Unique solution case (117.8 K, pdf, 03 Mar 2012)Slides: Lab 5, Linear algebra (71.4 K, pdf, 09 Dec 2011)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 S2012 (5.2 K, html, 08 Apr 2012)html: Determinants 2008 (227.1 K, pdf, 03 Mar 2012)Slides: Determinants, Cramers rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)Manuscript: Matrix add, scalar multiply and matrix multiply (156.7 K, pdf, 03 Mar 2012)Slides: Digital photos, Maxwell's RGB separations, visualization of matrix add (170.2 K, pdf, 03 Mar 2012)Slides: Inverse matrix, frame sequence method (97.9 K, pdf, 03 Mar 2012)Slides: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)Slides: Rank, nullity and elimination (154.1 K, pdf, 03 Mar 2012)Slides: Base atom, atom, basis for linear DE (117.6 K, pdf, 03 Mar 2012)Slides: Orthogonality (124.8 K, pdf, 03 Mar 2012)Slides: Partial fraction theory (160.7 K, pdf, 03 Mar 2012)Slides: The pivot theorem and applications (189.2 K, pdf, 03 Mar 2012)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