LAST UPDATE: Sun Jul 18 09:49:04 MST 2009 Office Hours: Tuesday-Thursday, 13:50-15:20 ------------------------------------------------- Math. 464/514, Fall'10 - Homework Assignments ------------------------------------------------- Assignments based on G. Strang, LINEAR ALGEBRA AND ITS APPLICATIONS, 4th edition, ISBN 0-03-010567-6 (will post problems that are numbered differently from 3rd edition) ------------------------------------------------------------------------------- 1( 8/24) Sec. 1.2-1.4 Geometry and Gaussian elimination 2( 8/26) Sec. 1.5 Elementary triangular factors ------------------------- Set 1 (due 9/ 2): p.9, 1.2(1,3,8,9), p.15, 1.3(11, 24, 30), p.26, 1.4(4,24) ------------------------- 1.2.3 (add:)...Find a fourth equation that leaves us with no solution. 1.3.11 (4th) = 1.3.1 (3rd) 1.3.24 (4th) = 1.3.3 (3rd) 1.3.30 (4th) = 1.3.11 (3rd) 1.4.24 Which three matrices E21,E31,E32 put A into triangular form U? A=[ 1 1 0] and E32*E31*E21*A = U. [ 4 6 1] Multiply those E's to get one matrix M that does elimination: [-2 2 0] MA = U. ------------------------------------------------------------------------------- 3( 8/31) Sec. 1.6 Inverses 4( 9/ 2) Sec. 2.1 Vector Spaces Sec. 2.2 mXn systems 4th Ed. 3rd Ed. Set 2 (due 9/ 9): p.39, 1.5(4,9,11,13,18), (p.39) p.52, 1.6(10,11), (p.51) P.65, 1.R(4,5,12,13,18) (p.60) ------------------------------------------------------------------------------- 5( 9/ 7) Sec. 2.3 Linear Independence, Basis, Dimension 6( 9/ 9) Sec. 2.4 Fundamental subspaces ------------------------- Set 3 (due 9/16): p.73, 2.1(2,7,8) p.85, 2.2(5,6,7,8) NOTICE CHANGE!!! problem 6 replaces 4 p.98, 2.3(2,6,8,10) ------------------------- 3rd Ed. 2.1.2c: change b3 to b1 throughout. 2.2.5 4th = 2.2.6 3d + {part b: same for [1 2 3][u] = [1] [2 4 4][v] [4] [w] 2.2.(6,7,8) 4th = 2.2.(7,8,9) 3rd 2.3.2 Find the largest possible number of independent vectors among v1=[ 1], v2=[ 1], v3=[ 1], v4=[ 0], v5=[ 0], v6=[ 0] [-1] [ 0] [ 0] [ 1] [ 1] [ 0] [ 0] [-1] [ 0] [-1] [ 0] [ 1] [ 0] [ 0] [-1] [ 0] [-1] [-1] This number is the ???? of the space spanned by the v's. 2.3.6 Choose 3 independent columns of U. Then make two other choices. Do the same for A. You have found bases for which spaces? U=[2 3 4 1] and A=[2 3 4 1]. [0 6 7 0] [0 6 7 0] [0 0 0 9] [0 0 0 9] [0 0 0 0] [4 6 8 2] 2.3.8 If w1, w2, w3 are independent vectors, show that the sums v1=w2+w3, v2=w1+w3, and v3=w1+w2 are independent. (write c1*v1+c2*v2+c3*v3=0 in terms of the w's. Find and solve equations for the c's.) 2.3.10 Find two independent vectors on the plane x+2*y-3*z-t=0 in R4. Then find 3 independent vectors. Why not 4? This plane is the nullspace of what matrix? ------------------------------------------------------------------------------- 7( 9/14) Sec. 2.4 Fundamental subspaces 8( 9/16) Sec. 2.4 continued ------------------------- Set 4 (due 9/23): p. 110, 2.4(1,3,5, 6,14,16,17,18, 27,36) ------------------------- 3rd Ed. p. 99, 2.4(1,3,5, 7,15,17,18,19, missing, missing) 2.4.27 A is an mXn matrix of rank r. Suppose there are right-hand sides b for which Ax=b has no solution. (a) What inequalities (< or <=) must be true between m, n, and r? (b) How do you know that A'y=0 has a nonzero solution? (A' = A^T) 2.4.36 Without multiplying matrices, find bases for the row and column spaces of A = [1 2][3 0 3]. [4 5][1 1 2] [2 7] ------------------------------------------------------------------------------- 9( 9/21) Sec. 2.5 Network and Graph Theory 10( 9/23) Sec. 2.5 continued ------------------------- Set 5 (due 9/30): p. 124, 2.5(4,5,6,9,11,16,17,18,21) ------------------------- 3rd Ed. p. 113, 2.5(4,5,6,9,11,missing,missing,missing,missing) 2.5.16 If there is an edge between every pair of nodes (complete graph), how many edges are there? The graph has n nodes, and edges from a node to itself are not allowed. 2.5.17 For both graphs drawn below, verify Euler's formula: (# of nodes) - (# of edges) + (# of loops) = 1. O------O------O O-----O | | | / \ / \ | | | / \ / \ O------O------O O-----O-----O | | | \ / \ / | | | \ / \ / O------O------O O-----O 2.5.18 Multiply matrices to find A'A, and guess how its entries come from the graph: (a) The diagonal of A'A tells how many ... ... into each node. (b) The off-diagonals (which are -1 or 0) tell which pairs of nodes are ... 2.5.21 The adjacency matrix M of a graph has M_{ij} = 1 if nodes i and j are connected by an edge (otherwise M_{ij} = 0). For the graph in problem 6 with 6 edges and 4 nodes, write down M and also M^2. Why does (M^2)_{ij} count the number of 2-step paths from node i to node j? ------------------------------------------------------------------------------- 11( 9/28) Sec. 2.6 Linear Transformations Sec. 3.1 Orthogonal subspaces 12( 9/30) Sec. 3.1 "" "" Sec. 3.2 Inner products ------------------------- Set 6 (due 10/7): p. 133, 2.6(6,7,18,47) p. 148, 3.1(7,9,12,16) p. 157, 3.2(8,11) ------------------------- 3rd Ed. (* = missing) p. 125, 2.6(7,8,19,*) p. 141, 3.1(7,9,12,16) p. 150, 3.2(8,11) 2.6.47 The 4x4 Hadamard matrix is entirely +1 and -1: H = [ 1 1 1 1 ]. Find H^(-1) and write v = [7; 5; 3; 1] [ 1 -1 1 -1 ] (i.e. v is a column vector) [ 1 1 -1 -1 ] as a combination of the columns of H. [ 1 -1 -1 1 ] ------------------------------------------------------------------------------- 13(10/ 5) Sec. 3.3 Projection and Least Squares 14(10/ 7) Sec. 3.4 Orthogonalization and the Gram-Schmidt process ------------------------------------------------------------------------------- 15(10/12) Orthogonal Projections ------------------------- Set 7 (due 10/19) p. 170, 3.3(3*,6,11,12,15,18*,19,20*,24,25*,26*) p. 185, 3.4(2*,4*,5,6,9,13,15,18,25) (problems marked by * are for practise; do not turn in!) ------------------------- 3rd Ed. same problems (on pp. 162, 180 resp.) **(10/14-15) Fall Break ------------------------------------------------------------------------------- 16(10/19) Ch.4.1-3: Some results on determinants. 17(10/21) MIDTERM EXAM (Chapters 1-3) ------------------------------------------------------------------------------- 18(10/26) Sec. 4.4: More on determinants 19(10/28) Sec. 5.1 Eigenvalues and Eigenvectors Sec. 5.2 Diagonal forms ------------------------- Set 8 (due 11/ 4) p. 206, 4.2(1,4,12,13,19,21,29,30) p. 215, 4.3(3,6,34) p. 240, 5.1(5) p. 250, 5.2(4,6,8,13) ------------------------- 3rd Ed. same problems (on pp. 162, 180 resp.) 4.2.1 p.218: same, but assume A=1/2 4.2.4: By applying row ops. to produce an upper triangular U, compute det| 1 2 -2 0| and det| 2 -1 0 0| | 2 3 -4 1| |-1 2 -1 0| | -1 -2 0 2| | 0 -1 2 -1| | 0 2 5 3| | 0 0 -1 -2| Exchange rows 3 and 4 of the second matrix and recompute the pivots and det. 4.2.12 Use row ops. to verify det| 1 a a^2| = (b-a)(c-a)(c-b) | 1 b b^2| | 1 c c^2| 4.2.13 = 4.2.11 in 3rd 4.2.19 = 4.2.17 in 3rd. 4.2.21 The inverse of a 2x2 matrix seems to have det = 1: detA^-1 = det( 1 | d -b|) ad-bc -----| | = ----- = 1 ad-bc|-c a| ad-bc What is wrong with this calculation? What is the correct detA^-1 ? 4.2.29 What is wrong with this proof that projection matrices have detP=1? P = A (A'A)^-1 A' so detP = detA( 1 ) detA' = 1. ---------- detA' detA 4.2.30 Show that the partial derivatives of ln(detA) give A^-1: f(a,b,c,d) = ln(ad-bc) leads to A^-1 = / Df/Da Df/Dc \ \ Df/Db Df/Dd / (where Df/Da means (partial) f / (partial) a ). 4.3.(3,6): same 4.3.34 With 2x2 blocks, you cannot always use block determinants! | A B | = |A||D| but | A B | ~= |A||D| - |B||C| | 0 D | | C D | (a Why is the first statement true?) Somehow B does not play! (b) Show by example that equality fails when C enters. (c) Show by example that the answer det(AD-BC) is also wrong. 5.1.(5,6,8,13): same ------------------------------------------------------------------------------- 20(11/ 2) Sec. 5.3 Difference equations and A^k 21(11/ 4) Sec. 5.4 Differential equations and exp(At) ------------------------- Set 9 (due 11/11) p. 262, 5.3(4,8,16,19) p. 275, 5.4(1,4,5) ------------------------- 3rd Ed. same problems (on pp. 162, 180 resp.) p. 272, 5.3(2*,7,13,16) 5.3.4: in 5.3.2 put G1 =1 and show Gn approaches 2/3. Set up the iteration matrix A, find its eigenvalues and eigenvectors and find the limit of A^n by using the diagonal form. p. 286, 5.4(1,4,5) ------------------------------------------------------------------------------- 22(11/ 9) Complex matrices 23(11/11) Sec. 5.6 Similarity ------------------------- Set 11 (due 11/23) p. 288, 5.5(7,8,11,19) p. 302, 5.6(4,7,13,24,28,29) p. 307 review 5.(8,11,20) ------------------------- 3rd Ed. same (on pp. 301, 315 & 319) except: 5.6.28-29 in 4th ==> 5.6.29 in 3rd Ed. ------------------------------------------------------------------------------- 24(11/16) Sec. 5.6 Generalized eigenspaces and the Jordan canonical form. 25(11/18) Sec. 6.1 Extrema and Quadratic Forms, Sec. 6.2 Positive Definite Matrices ------------------------------------------------------------------------------- 26(11/23) Sec. 6.2 cont.: Symmetric Matrices and the Law of Inertia ------------------------- Set 12 (due 12/ 2) p.302, 5.6(39--hint: see prob.4, Appendix B), p.427, App.B (1) p.316, 6.1(4,7,17) p.326, 6.2(7,9,11,14,18,19,27,28) ------------------------- 3rd Ed. same except: 5.6.39 Prove in 3 steps that A' is always similar to A (we know that the evals. are the same, the evecs are the problem): (a) For A = 1 block, find Mi = permutation so that Mi^-1 Ji Mi = Ji' (b) For A = any J, build M0 from blocks so that Mo^-1 J M0 = J' (c) For any A = M J M^-1: show that A' is similar to J' and so to J and to A. 6.1.17 If A has independent columns, then A'A is square and symmetric and invertible (Sec. 4.4). Rewrite x'A'Ax to show why it is positive except when x=0. The A'A is pos.def. (PD). 6.2.7: same except use A = [10 6] [ 6 10]. 6.2.14,18 4th == 6.2.15,19 3rd 6.2.19 Which 3x3 symmetric matrices produce these functions f(x) x'Ax? (a) f = 2(x1^2 + x2^2 + x3^2 - x1 x2 - x2 x3) (b) f = 2(x1^2 + x2^2 + x3^2 - x1 x2 - x1 x3 - x2 x3) 6.2.27 With pos. pivots in D, the factorization A = L D L' becomes L B B L' (B = sqrt(L)). Then C = L B gives the Cholesky factorization A = C C', which is "symmetrized LU". From C = [3 0] find A. From A = [4 8] find C. [1 2] [8 25] 6.2.28 In the Cholesky factorization A = C C', with C = L B (B as above) the square roots of the pivots are in the diagonal of C. Find C (lower triangular) for A = [9 0 0] and A = [1 1 1] [0 1 2] [1 2 2] [0 2 8] [1 2 7] **(11/25-28) Thanksgiving Break ------------------------------------------------------------------------------- 27(11/30) Sec. 6.2 cont.: Symmetric Matrices and the Law of Inertia Sec. 6.3 The Singular Value Decomposition: 28(12/ 2) Pseudoinverses ------------------------------------------------------------------------------- 29(12/ 7) Exam 2 (covers from 4.1 to 6.2 - one page of notes OK) 30(12/ 9) Pseudoinverses and other applications of the SVD ------------------------------------------------------------------------------- ----------- Final Examination: Thursday, December 14 7:30--9:30 in XXX??? (Cummulative - Open book+notes) -------------------------------------------------------------------------------