TopicsSections 3.4, 3.5, 3.6 The textbook topics, definitions, examples 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)

Review of MatricesVector and vector operations. 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. Theorem 1. Matrix algebra A+B=B+A A+(B+C)=(A+B)+C A(BC)=(AB)C A(B+C)=AB+AC, (D+E)F=DF+EFSpecial matricesdiagonal matrix upper and lower triangular matrices square matrix Def: Zero matrix 0 Theorems: 0+A=A+0, A0=0, 0A=0 Def: Identity matrix I Theorems: AB=AC ==> B=C is false AI=A, IB=B Def: Augmented matrix of vectors [v1 v2 v3 ... vn] Same as Maple v:=for n=3 Theorems: Ax in terms of columns of A AB in terms of columns of B Def: Matrix A has inverse matrix B means AB=BA=I Inverse matrix Definition: A has an inverse B if and only if AB=BA=I. Theorem 1. An inverse is unique. THEOREM 1a. If A has an inverse, then A is square. Non-square matrices don't have an inverse. THEOREM 1b. The zero matrix does not have an inverse. Theorem 1. The inverse of a matrix is unique. When it exists, then write B = A^(-1) Theorem 2. Inverse of a 2x2 matrix A:=Matrix([[a,b],[c,d]]) equals B=(1/det(A))Matrix([[d,-b],[-c,a]]) Theorem 3. (a) (A^(-1))^(-1)=A (b) (A^n)^(-1)=(A^(-1))^n for integer n>=0 (c) (AB)^(-1)=B^(-1) A^(-1) Theorem 4. The nxn system Ax=b with A invertible has unique solution x=A^(-1)b 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 to be answered:What did I just do, by finding rref(A)?You solved the system Ax=0 by finding the Last Frame in a combo, swap, mult sequence starting with matrix A or augmented matrix aug(A,0). The RREF is the Last Frame. From it, use the Last Frame Algorithm to find the general solution in scalar form, then in vector form. Problems 3.4-17 to 3.4-22 are homogeneous systems Ax=0 with matrix A in reduced echelon form. Apply the last frame algorithm then write the general solution in vector form. EXAMPLE. A 3x3 matrix. Special Solutions of Gilbert Strang. How to find the vector general solution. How to find x_p: Set all free variable symbols to zero How to find x_h: Take all linear combinations of the Special Solutions. Superposition: x = x_p + x_h = General Solution Def: Elementary Matrix. It is constructed from the identity matrix I by applying exactly one operation combo, swap or mult. Conventions E = combo(I,s,t,c) [E is an elementary combo matrix] E = swap(I,s,t) [E is an elementary swap matrix] E = mult(I,t,m) [E is an elementary multiply matrix] EXAMPLE: For the 2x2 identity I=Matrix([[1,0],[0,1]]), E=mult(I,2,m)=Matrix([[1,0],[0,m]]) Definitions and details:: 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 sequence. Symbols E_{n}, E_{n-1}, ... E_{1}are elementary square matrices that represent the operations combo, swap, mult that created the sequence.

How to compute the inverse matrixDef: AB=BA=I means B is the inverse of A.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 sequence of frames augment(A,I) then toolkit operations combo, swap, mult to arrive at the Last Frame augment(I,B) The inverse of A equals matrix B in the right panel (last frame).: Inverse matrix, frame sequence method (97.9 K, pdf, 03 Mar 2012)Slides: Matrix add, scalar multiply and matrix multiply (156.7 K, pdf, 03 Mar 2012)SlidesElementary matrices.How to write a combo-swap-mult sequence as a matrix product Fundamental theorems on combo-swap-mult sequencesTHEOREM. If B immediately follows A in a combo-swap-mult sequence, then B = E A, where E is an elementary matrix having EXACTLY ONE of the following forms: E=combo(I,s,t,c), or E= swap(I,s,t), or E= mult(I,t,m) Proof: See problem 3.5-39.THEOREM. If a combo-swap-mult sequence starts with matrix A and ends with matrix 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. Every elementary matrix E has an inverse. It is found as follows: Elementary Matrix Inverse Matrix E=combo(I,s,t,c) E^(-1)=combo(I,s,t,-c) E=swap(I,s,t) E^(-1)=swap(I,s,t) E=mult(I,t,m) E^(-1)=mult(I,t,1/m)Web References: Elementary matrices: Elementary matrix theorems (161.5 K, pdf, 03 Mar 2012)SlidesInverses of elementary matrices.PROBLEM. Solve B=E3 E2 E1 A for matrix A. ANSWER. 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.

The textbook topics, 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)College Algebra Background:College algebra determinant definition Sarrus' rule for 2x2 and 3x3 matrices.References for 3.6 determinant theory and Cramer's Rule: Determinants 2008 (227.1 K, pdf, 03 Mar 2012)Slides: Determinants, Cramer's rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012) How to do 3.5-16 in maple. combo:=(A,s,t,c)->linalg[addrow](A,s,t,c); A:=Matrix([[1,-3,-3],[-1,1,2],[2,-3,-3]]); B:=A^(-1); # expected answer id:=<1,0,0|0,1,0|0,0,1>; A1:= < A | id >; linalg[rref](A1); # inverse in right panel A2:=combo(A1,1,2,1); # Doing the steps one at a time A3:=combo(A2,1,3,-2); WARNING. Some linalg functions return a matrix X with wrong format. Be willing to use XX:=convert(X,Matrix), as needed. Or, retire linalg and use only the LinearAlgebra package. See problem notes chapter 3Manuscript: Problem notes S2013 (5.1 K, html, 06 Dec 2012)htmlMethods 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=mult(A,t,m), then |A| = (1/m) |B|Swap rule: B=swap(A,s,t), then |A| = (-1) |B|Combo rule: B=combo(A,s,t,c), then |A| = |B|

Exam 1 review, questions and examples on exam problems 1,2,3,4,5. Time permitting: Maple Lab 3, problems 1,2,3 details, due after Exam 1.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)

\Survey of Main theorems:Computation by the 4 rules, cofactor expansion, hybrid methods. Determinant product theorem det(AB)=det(A)det(B). Cramer's Rule for solving Ax=b:

x_{1}= delta_{1}/delta, ... , x_{n}= delta_{n}/delta Adjugate formula: A adj(A) = adj(A) A = det(A) I Adjugate inverse formula inverse(A) = adjugate(A)/det(A).Results on DeterminantsExamples: Computing det(A) easily. When does det(A)=0?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).Discussion of 3.5 problems. Lecture Ideas of rank, nullity, dimension in examples.: Rank, nullity and elimination (156.3 K, pdf, 21 Dec 2012) More on Rank, Nullity dimension 3 possibilities elimination algorithmSlidesLectureCofactor 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. Hybrid methods to evaluate det(A). 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 Cramer's rule. How to form the matrix of cofactors and its transpose. The adjugate matrix.THEOREM.The 4 rules for computing any determinant can be compressed into two rules, det(triangular matrix)=the product of the diagonal elements, det(EA)=det(E)det(A), where E is an elementary matrix, combo, swap or mult.Determinant product theoremdet(AB)=det(A)det(B) for any two square matrices A,B Proof details. Example.THEOREM. The adjugate formula A adj(A) = adj(A) A = det(A) I.THEOREM. Adjugate inverse formula: inverse(A) = adj(A)/det(A).: Determinants 2010 (227.1 K, pdf, 03 Mar 2012)Slides: Determinants, Cramer's rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)Manuscript: Problem notes S2013 (5.1 K, html, 06 Dec 2012)htmlTranspose matrix(A^T)^T = A (A + B)^T = A^T + B^T (AB)^T = B^T A^T det(A)=det(A^T)Further properties of the adjugate matrixComputing det(A) from A and adj(A) in 10 seconds Problems involving adj(A): examples from exams. Adjugate identity A adj(A) = adj(A) A = det(A) I 3x3 case: 6 ways to compute det(A) from A, adj(A). 3x3 case: the 6 cofactor expansionsReferences for chapters 3 and 4, Linear Algebra: Linear algebraic equations, no matrices (460.2 K, pdf, 10 Feb 2012)Manuscript: Matrix Equations (292.1 K, pdf, 07 Feb 2012)Manuscript: vector models and vector spaces (144.1 K, pdf, 03 Mar 2012)Slides: Vectors and Matrices (426.4 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 [exam review and maple lab 3] (184.6 K, jpg, 08 Feb 2008)Transparency: Elementary matrix theorems (161.5 K, pdf, 03 Mar 2012)Slides: Linear equations, reduced echelon, three rules (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: Lab 5, Linear algebra (170.0 K, pdf, 04 Dec 2012)Maple: Problem notes S2013 (5.1 K, html, 06 Dec 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 (156.3 K, pdf, 21 Dec 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