Thu Jan 15: First day of class. We went over the syllabus. We covered parts of chapters 1.1 and 1.2 from the the textbook. You are responsible for reading all of 1.1 and 1.2, and for solving HW 1.1 #5,#9,#17+18(just explain how to do it), 23, 29, 31, 33, 34, and HW 1.2 #1, 3, 5, 7, 9, 13, 15, 21, 23, 25, 27. We will turn in the quiz on Tuesday as a take-home, though most quizzes are due in class. You can check your work against the answer key.
Fri-Mon: Make sure to reread 1.1 and 1.2, work on the homework, and skim 1.3.
Tue Jan 20: We went over 1.3 including stoichiometry. For homework, reread 1.3 and solve HW1.3 #5, 7, 9, 11, 13, 22, 29. You can omit the calculations on 13 and 29, but make sure you know what calculations you would do. We worked on a worksheet and quiz. You can check your work against the answer key.
Thu Jan 22: We went over spans (1.3), reviewed stoichiometry (1.6), and covered 1.4 (matrix-vector multiplication). For homework, reread 1.4 and solve HW1.4 #1,3,13,21,25. We worked on a worksheet and quiz. You can check your work against the answer key.
Tue Jan 27: We will go over homogenous and particular solutions and parametric forms of solutions (1.5). For homework, reread 1.5 and be able to solve HW1.5 #7, 11, 19, and 5 versus 17. We worked on a quiz. You can check your work against the answer key.
Thu Jan 29: First exam. There is a practice exam with sample answers. The real exam and key are also available.
Tue Feb 3: We covered 1.7 linear independence and introduced 1.8 linear transformations. HW1.7 #1, 3, 11, 15, 17, 19, 31. Reread section 1.7 and 1.8. Skim 1.9. The quiz is available, along with a worked version with some answers to in-class questions.
Thu Feb 5: We covered 1.8 and 1.9, matrices of linear transformations. HW1.9#1,15,17 (and Mathy-majors do #34). Skim 2.1. The quiz is available, along with a worked version with HW1.7#13.
Tue Feb 10: 2.1 Matrices are vectors too! Using the idea of 1.8, matrices are functions and so we can add and subtract them (if their domains and codomains agree) and multiply them by a scalar. We can even compose them. Using the ideas of 1.9 we can find the matrices of these conglomerate functions. HW 2.1 #1,3,5,7,9,10,11,12,19,21 and for later #27,28,40. The quiz is available, along with a worked version.
Thu Feb 12: 2.2 and 2.3 inverse matrices. In class we talked about left inverses to gain a little more familiarity with rectangular matrices, but mostly so we appreciated how much peace of mind came from thinking about square matrices. We spent some time with diagonal matrices as well. They are very important and we'll base a lot of our understanding of linear algebra on them. HW 2.2 #1,11,13,14,15,17,21. HW 2.3 #1,3,5,13,18 (can #7 be done easily?). The quiz is available, along with a worked version.
Tue Feb 17: SNOW DAY
Thu Feb 19: SNOW DAY
Tue Feb 24: Quick review of 1.7, 1.8, 1.9, 2.1, 2.2, 2.3. 5 minutes to breath. Then the exam over those sections. We will not cover chapter 3. The practice exam, a worked version, and a worked version with challenges are available. You can also check the real exam and a worked version.
Thu Feb 26: Chapter 4.1 -- we started getting used to weirder vector spaces: specifically, function spaces. We talked about subspaces as a cheap way to get more useful vector spaces without starting from scratch. The homework was 4.1 #1-8, 9, 11, 13, 15, 17, 19. Show the solutions to y" = y form a subspace. The quiz is available, along with answers to the quiz and some additional answers.
Tue Mar 3: Chapter 4.2 -- a standard form for subspaces (Col(A) and Nul(A)). HW 4.2 #1,3,5,7,9,11,13,31. The quiz and a worked version are available along with the long handout on technical math words (written during my fever).
Thu Mar 5: SNOW DAY
Tue Mar 10: Chapter 4.3 and 4.3 -- a very standard form for subspaces: bases and coordinates. The quiz is available along with a worked version.
Thu Mar 12: Exam (worked version) over column and null spaces. A practice exam is now available along with a worked version and another worked version. I also wrote out how to do 3b with RREF, but I seriously doubt anyone would do it that way on the exam.
Tue Mar 24: Chapter 5: Eigenvalues and eigenvectors. Turning matrices into numbers. The quiz is available, along with a worked version.
Thu Mar 26: 5.2 and 5.3. Finding eigenvectors from an eigenvalue, finding eigenvalues from a very small matrix, doing useful things with them. The quiz is available, along with a worked version.
Tue Mar 31: Review. (1) Finding an eigenvector given matrix and eigenvalue, (2) finding an eigenvalue given matrix and eigenvector, (3) finding both eigenvalue and eigenvector given only matrix. Worked fibonacci example. The practice exam is available, along with a worked version.
Thu Apr 2: The chapter 5 exam and a worked version are available.
Tue Apr 7: Chapter 6.1 and 6.2: the standard inner product, orthogonal vectors, and coordinates in an orthogonal basis. Also called dot products or matrix multiplication. The beginning of the big easy. The quiz is available, along with a worked version.
Thu Apr 9: Chapter 6.3 through 6.5: orthogonal projections and least squares. We know how to find the coordinates of a vector in an orthogonal basis, so what do we do if we just have a few orthogonal vectors, not an entire basis? Answer: The best we can. The quiz and a worked version are available.
Tue Apr 14: Chapter 6 cleanup and review. Clearly explain the advantage of the "G" frame of reference. Make sure everyone can do a big problem. The worksheet is available. The practice exam is currently very sketchy.
Thu Apr 16: Chapter 6 review and an intro to Chapter 7.1.
Tue Apr 21: Chapter 6 exam. I made another practice exam with slightly better numbers, and of the worked version. The real exam and a worked version are also available.
Tue Apr 28: Chapter 7.1-7.3: principal axes to solve energy problems
and least squares. Here is the worksheet and a worked version. You can ask a calculator (or this one) to
find the square roots and eigenpairs for you.
format short g
A = [3.0224,1.6592,-4.888;1.6592,4.1136,8.296;-4.888,8.296,58.56]
B = [1,1,1;1,2,3;1,3,14]
b = chol(B)
[g,d] = eigs(b'\A/b)
w = 10*g(:,1)
v = b\w
v'*B*v
v'*A*v
You can also use fewer commands to get the same answer as
explained here,
but I think the explanation is not as simple (and the floating
point operation count is the same; or larger if matlab doesn't
notice it can use Cholesky).
format short g
A = [3.0224,1.6592,-4.888;1.6592,4.1136,8.296;-4.888,8.296,58.56];
B = [1,1,1;1,2,3;1,3,14];
[v,d] = eigs( B\A, 1 );
v = v*sqrt(100/(v'*B*v))
v'*B*v
v'*A*v
I should also mention, this problem is so important that matlab does have
special purpose code to solve it. [v,d]=eigs(A,B,1,"la") sets
v to be the vector that maximizes the ratio d =
(v'*A*v)/(v'*B*v), as long as A and B are real
symmetric matrices. The "la" indicates we want the largest ratio,
not just the largest absolute value of the ratio (the default). You can see the
difference if you use A = [1.5968 2.1344 -12.016;2.1344 3.9552 10.672;-12.016 10.672 22.92].
Thu Apr 30: Chapter 7.4/7.5: spectral decomposition for rectangular matrices, and low rank approximation.
format short g A=[ 11.658 11.65 11.658 34.886 23.276; 6.639 6.628 20.765 34.082 13.313; 7.927 7.969 11.075 26.971 15.874; 3.411 3.41 -4.419 2.392 6.847; 24.004 23.999 3.602 51.635 48.013; 8.782 8.777 26.036 43.585 17.549] [U,S,V] = svd(A); U=U.*(abs(U)>0.0001); V=V.*(abs(V)>0.0001); U,S,V u1 = U(:,1); u2 = U(:,2); v1 = V(:,1); v2 = V(:,2); s1 = S(1,1); s2 = S(2,2); x=rand(5,1);b=A*x; y = v1*(u1'*b)/s1 + v2*(u2'*b)/s2; [ b, A*y ]
May 7th: Final exam. Here is a practice exam and a worked version. Here are some variations on page 2. Here is a variation on page 3. The final exam and a worked version are also available.