TopicsThe textbook definitions and theorems

Edwards-Penney 3.1, 3.2, 3.3 (3.0 K, txt, 07 Feb 2013)

Edwards-Penney 3.4, 3.5, 3.6 (4.0 K, txt, 04 Feb 2013)

Linear Algebraic Equations sections 3.1, 3.2Frame sequence defined. Theater 35mm movies project at 30framesper second.

A 35mm filmstrip (7.0 K, jpg, 12 Dec 2012) Streaming video provider NetFlix usesframesto make a filmstrip, each frame an image thumbnail of a DVD cover.

A NetFlix Wii filmstrip (19.5 K, jpg, 12 Dec 2012) Solutions to linear algebra problems can use a similar filmstrip scheme, with the NetFlix DVD cover replaced by an image of a completed solution step, called aframe.

A filmstrip of 3 solution steps, called a frame sequence (68.3 K, pdf, 12 Dec 2012) Aframe sequencerecords the steps needed to solve a system of equations. The steps are determined by a basic set of operations on equations (the Toolkit). Toolkit: combination, swap, multiply Briefly: combo, swap, mult Plane and space geometry: the meaning of simultaneous equations The three possibilities Unique solution No solution Infinitely many solutions Method of eliminationExample: unique solution x + 2y = 1 x - y = -2Example: no solution x + 2y = 1 x + 2y = 2Example: infinitely many solutions x + 2y = 1 0 = 0 References:: Intro Linear Equations, Toolkit, 3 Possibilities, Matrices (237.3 K, pdf, 15 Dec 2012)Slides: Unique solution case (110.1 K, pdf, 14 Dec 2012)Slides: No solution case (79.7 K, pdf, 03 Mar 2012)Slides: Infinitely many solutions case (131.1 K, pdf, 03 Mar 2012)Slides: Linear algebraic equations, no matrices (460.2 K, pdf, 10 Feb 2012) Parameters in the general solution Calculus: Parametric equation of a line or a plane. Differential equations example, problem 3.1-26 y'' -121y = 0, y(0)=44, y'(0)=22 General solution given: y=A exp(11 x) + B exp(-11 x) Substitute y into y(0)=44, y'(0)=22 to obtain a 2x2 system for unknowns A,B that has the unique solution A=23, B=21.ManuscriptPrepare 3.1 problems for next collection.See problem notes section 3.1:: Problem notes S2013 (5.1 K, html, 06 Dec 2012)htmlReferences for ER-1, ER-2 due next Week: Problems ER-1, ER-2 (109.6 K, pdf, 04 Dec 2012)Exam Review: Sample solution ER-1. (184.6 K, jpg, 08 Feb 2008)TransparencyMore References: Ch3 Page 149+, Exercises 3.1 to 3.6 (869.6 K, pdf, 25 Sep 2003)Transparencies: 3x3 Frame sequence and general solution (315.9 K, jpg, 12 Dec 2012)Transparency

Maple lab 2 problem 1Discussion: Option 1: Freezing pipes maple lab 2 Problem: u' + ku = kA(t) Integration methods Tables Maple Answer check by computer Links for maple lab 2:: Maple Lab 2, Newton cooling freezing pipes (240.5 K, pdf, 04 Dec 2012)Option 1: maple worksheet text freezing pipes (1.3 K, txt, 15 Mar 2012)Option 1: Maple Lab 2, Newton cooling swamp cooler (240.5 K, pdf, 04 Dec 2012)Option 2: maple worksheet text swamp cooler (1.3 K, txt, 15 Mar 2012)Option 2

For more on superposition y=y_p + y_h, see Theorem 2 in the link

Linear DE part I (152.7 K, pdf, 07 Aug 2009)

For more about home heating models, read the following links.: Linear equation applications, brine tanks, home heating (374.2 K, pdf, 28 Jul 2009)Manuscript: Brink tanks (95.3 K, pdf, 04 Mar 2012)Slides: Home heating (109.8 K, pdf, 04 Mar 2012)SlidesLecture: 3.1, 3.2, 3.3TopicsFrame sequence Toolkit: combo, swap, multiply [EP 3.2] Plane and space geometry The three possibilities [EP 3.1] Unique solution No solution Infinitely many solutionsDefinitionsLead variable Free variable Signal equation Echelon form [EP 3.2] The last frame test [EP 3.3] The last frame algorithm [EP 3.2] A detailed account of the three possibilities Unique solution == zero free variables No solution == signal equation Infinitely many solutions == one+ free variablesHow to solve a linear systemToolkit: swap, combo, mult Toolkit operations neither create nor destroy solutions! Frame sequence examples Computer algebra systems and error-free frame sequences. How to use maple to make a frame sequence without errors.: Utah Maple Tutorial (2.0 K, html, 03 Dec 2012)HelpSolved ProblemsExample 4 in 3.2A:=Matrix([ [1,-2,3,2, 1], [0, 0,1,0, 2], [0, 0,0,1,-4]]); b:=<10,-3,7>; x1 - 2x2 + 3x3 + 2x4 + x5 = 10, x3 + 2x5 = -3, x4 - 4x5 = 7 ==> A is upper triangular; system is upper triangular ==> Solve by 3.2 Algorithm Back Substitution. Back-substitution.Warning: The naming convention in the textbook [EP 3.2] differs from a substantial number of references.

Remarks on back-substitution 2013 (1.4 K, txt, 14 Dec 2012) Back-Substitution, be it the textbook algorithm or another, should be presented as combo operations in frames. Please avoid isolated, incomplete algebraic gibberish.

A gibberish example (1.2 K, txt, 13 Dec 2012)DEFINITIONTheleading variablein a equation is the first variable, in variable list order, with a nonzero coefficient.DEFINITIONA frame isechelonprovided (1) Each nonzero equation has a leading variable different from all other equations. (2) All variables preceding a leading variable, in variable list order, are absent from all following equations.DEFINITIONIn anEchelon Systemthepivotsare the coefficients of the leading variables, in turn calledpivot variables.MATLAB EXPERTS Back-Substitution ExampleAssume U is an nxm echelon (upper triangular) matrix with nonzero diagonal entries U(i,i). Can you decode the following back substitution algorithm for Ux=b, written in Matlab syntax? n = length(b); x = zeros(n,1); for i=n:-1:1 x(i) = (b(i)-U(i,:)*x)/U(i,i); end NON-EXPERTS. The answer is expressed in terms of dot-products, a COMBO operation.Problem 3.2-24The book's answer is wrong, it should involve k-4. See references on 3 possibilities with symbol k.: 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: Examples 11, 12 in Linear algebraic equations no matrices (460.2 K, pdf, 10 Feb 2012)Answer checks should use the online FAQ.Manuscript: Problem notes S2013 (5.1 K, html, 06 Dec 2012)?> In solved problems submitted for grading, please use frames to display the solution, as in lecture examples. Expected is a sequence of augmented matrices. Maple may be used to make the frames. A maple last frame answer check is linalg[rref](A). Browse the Linear Algebra help in the on-line tutorial:html: Utah Maple Tutorial (2.0 K, html, 03 Dec 2012)HelpReferences for this lecture.: Intro Linear Equations, Toolkit, 3 Possibilities, Matrices (237.3 K, pdf, 15 Dec 2012)Slides: Unique solution case (110.1 K, pdf, 14 Dec 2012)Slides: No solution case (79.7 K, pdf, 03 Mar 2012)Slides: Infinitely many solutions case (131.1 K, pdf, 03 Mar 2012)Slides: Linear algebraic equations, no matrices (460.2 K, pdf, 10 Feb 2012)Manuscript

TopicsThe textbook definitions and theorems

Edwards-Penney 3.1, 3.2, 3.3 (3.0 K, txt, 07 Feb 2013)

Edwards-Penney 3.4, 3.5, 3.6 (4.0 K, txt, 04 Feb 2013)Project: Switch from equations to matrices, which is the interface for a computer algebra system [maple, mathematica] or a numerical workbench [matlab, scilab]. Translation of equation models [EP 3.4] Equality of vectors Scalar equations to augmented matrix Augmented matrix to scalar equations Matrix toolkit: Combo, swap and multiply Frame sequences for matrix models. Last frame test for a matrix. The RREF of a matrix. Last frame algorithm for a matrix. Scalar form of the solution. Vector form of the solution.THEOREMSin 3.3 Theorem 1. RREF is unique Theorem 2. Three possibilities Theorem 3. More variables than equations ==> infinitely many Theorem 4. Ax=0 has unique solution x=0 <==> rref(A)=I Vector add and scalar multiply: componentwise Matrix add. Matrix scalar multiply. Matrix multiplication on paper. Digital photographs. Matrix add and scalar multiply. Maxwell's RGB separation. Photoshop operations and matrix algebra. 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)Maple Text: Frame Sequence in maple, Exercise 3.2-14 (3.0 K, txt, 10 Feb 2012) Answer checks should use the online FAQ.Maple Text: Problem notes S2013 (5.1 K, html, 06 Dec 2012) References:html: Linear algebraic equations, no matrices (460.2 K, pdf, 10 Feb 2012)Manuscript: Intro Linear Equations, Toolkit, 3 Possibilities, Matrices (237.3 K, pdf, 15 Dec 2012)Slides: Matrix Equations (292.1 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: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)Slides

Exam 1 review, questions and examples on exam problems 1,2,3,4,5. Time permitting: Maple Lab 2, problems 1,2,3 details, due week 6.Sample Exam: Exam 1 key from S2012. See also F2010, exam 1.

Exams and exam keys for the last 5 years (19.6 K, html, 30 Apr 2013)

TopicsThe textbook definitions and theorems

Edwards-Penney 3.1, 3.2, 3.3 (3.0 K, txt, 07 Feb 2013)

Edwards-Penney 3.4, 3.5, 3.6 (4.0 K, txt, 04 Feb 2013) The three possibilities The Toolkit: combo, swap, mult Frame sequence Last frame test for equations. Reduced echelon system. 1. Each nonzero equation has a leading variable with leading coefficient one. 2. Zero equations are last 3. Lead variables are maintained in variable list order Last frame test for matrices. Row reduced echelon form (RREF). 1. Each nonzero row has a leading one. 2. Zero rows are last 3. Leading ones are in columns of the identity matrix, called pivot columns. These columns match the order of initial columns of the identity matrix, but final columns may be missing. Last frame algorithm 1. Apply the last frame test. Proceed only if the system passes the test. If in matrix form, then convert to a scalar system. 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.Example1. The unique solution case C:=Matrix([[1,2,1,1],[1,3,1,2],[1,1,2,3]]); x + 2y + z = 1 x + 3y + z = 2 x + y + 2z = 3 Scalar form of the unique solution. Why only lead variables in the last frame? Why no free variables?Example2. The no solution case C:=Matrix([[1,2,1,1],[1,3,1,2],[2,5,2,4]]); x + 2y + z = 1 x + 3y + z = 2 2x + 5y + 2z = 4 We will find a signal equationExample3. The infinitely many solution case C:=Matrix([[1,2,1,1],[1,3,1,2],[2,5,2,3]]); x + 2y + z = 1 x + 3y + z = 2 2x + 5y + 2z = 3 Find the last frame How many free variables? Count is called the RANK. Apply the last frame algorithmExample4. The infinitely many solution case C:=Matrix([[1,2,1,1],[3,6,3,3],[0,0,0,0]]); x + 2y + z = 1 3x + 6y + 3z = 3 0 = 0 Find the last frame How many free variables? Apply the last frame algorithmExample5. The infinitely many solution case C:=Matrix([[0,0,0,0],[0,0,0,0],[0,0,0,0]]); 0 = 0 0 = 0 0 = 0 Find the last frame How many free variables? Apply the last frame algorithm The three possibilities with symbol k. x + ky = 2, (2-k)x + y = 3. Examples with symbols.: Three possibilities, theorems on infinitely many solutions, equations with symbols (129.0 K, pdf, 03 Mar 2012)Slides: 3 possibilities with symbol k (98.8 K, pdf, 03 Mar 2012)Slides: 3 possibilities with symbol k (60.0 K, pdf, 31 Jan 2010)Beamer slides: Examples 11, 12 in Linear algebraic equations no matrices (460.2 K, pdf, 10 Feb 2012) Reading:Manuscript: Linear algebraic equations, no matrices (460.2 K, pdf, 10 Feb 2012)Manuscript: Intro Linear Equations, Toolkit, 3 Possibilities, Matrices (237.3 K, pdf, 15 Dec 2012)Slides: Unique solution case (110.1 K, pdf, 14 Dec 2012)Slides: No solution case (79.7 K, pdf, 03 Mar 2012)Slides: Infinitely many solutions case (131.1 K, pdf, 03 Mar 2012)Slides: Matrix Equations (292.1 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: More on digital photos, checkerboard analogy (141.9 K, pdf, 03 Mar 2012)Slides: Inverse matrix, frame sequence method (97.9 K, pdf, 03 Mar 2012)Slides

Chapters 3 and 4: Last frame algorithm, Elimination, Rank, Nullity (156.3 K, pdf, 21 Dec 2012)Slides: Elementary matrix theorems (161.5 K, pdf, 03 Mar 2012)Slides: Three possibilities, theorems on infinitely many solutions, equations with symbols (129.0 K, pdf, 03 Mar 2012)Slides: Problem notes S2013 (5.1 K, html, 06 Dec 2012)html: Ch3 Page 149+, Exercises 3.1 to 3.6 (869.6 K, pdf, 25 Sep 2003)Transparencies: 3x3 Frame sequence and general solution (315.9 K, jpg, 12 Dec 2012)Transparency: Lab 5, Linear algebra (170.0 K, pdf, 04 Dec 2012)Maple: Vectors and Matrices (426.4 K, pdf, 07 Feb 2012)Manuscript: vector models and vector spaces (144.1 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 (1.9 K, txt, 04 Apr 2013)Text: Problem notes S2013 (5.1 K, html, 06 Dec 2012)html