Instructor: Prof. David Nadler
GSI:
Email:
Office: 853 Evans Hall
Office Hours: Monday 4-6pm, Thursday 4-5pm. All office hours will be held in 853 Evans Hall.
Content: From the online schedule: Matrices, vector spaces, linear transformations, inner products, determinants. Eigenvectors. QR factorization. Quadratic forms and Rayleigh's principle. Jordan canonical form, applications. Linear functionals.
Textbook: 'Linear Algebra Done Right', Axler (Springer, Second Edition)
Course Website: Additional course information, including the grading policy for the course and the course syllabus. Homework is posted here (and below).
Exams: In-class midterm during lecture meeting: Thursday, October 24 2013, Material: TBA
Final Exam: Wednesday, December 18 2013, 3-6pm (Exam Group 11)
piazza.com There is a class forum at piazza.com, if you would like to added to this then please send me an email. Use this forum to ask any questions you have concerning the material covered during class and on any problems you have with homework; also, feel free to answer any questions you feel comfortable discussing with your fellow students. Try to be civil with each other!
Notes: I will post any (hopefully?) helpful notes I prepare and hand out during the semester here.
Extras: Here is some useful information on problem solving techniques given by the Hungarian mathematician George Polya.
Here are some useful notes written by Prof. George Bergman on basic mathematical language.
Here is some useful advice on how not to lose marks on exams by a former colleague Andrew Critch.
History: Here is an interesting article about the German mathematician Hermann Grassmann and his involvement in the development of linear algebra during the 19th Century.
Here is a preliminary review for the upcoming first midterm.
Here are some tips for how to approach some proofs.
Here is a note about matrices and linear maps.
Here is a short review for the final.
Here is a note about operators on a complex inner product space.
Here is a note about operators on a real inner product space. (Note: during discussion section I made a mistake in the diagram.)
Here is a note on on Jordan form. It indicates how to determine the Jordan form for operators on small dimensional spaces and provides an algorithm for computing the Jordan basis. Here are solutions to the exercises.
Homework is due Wednesdays at the beginning of the discussion section for which you are registered.
Late homework will not be accepted.
If you are unable to submit your homework at the required time then you can leave it outside my office (853 Evans Hall) at any time before it is due. Please email me if you intend to leave your homework in my mailbox.
Collaboration on homework is welcome and encouraged although if you are working with another student please state that you have done so (eg. if you work with E. Nother on a particular question just write "This question was completed with E. Nother."). However, all homework assignments must be written up individually. Failure to declare collaboration with another student will result in a grade penalty (and it is remarkably simple to tell when students have copied each other). Also, if you have used a textbook or online notes to help you understand/solve a problem please cite a reference (eg. if you used pages 52-60 of Prof. X's online lecture notes just write "This question used p.52-60 of Prof. X's online lecture notes, available at www.math.com/~profx/linalg)
Grading: Homework will be graded on a scale of 0-3 with 0=no effort, 1=effort with some progress, 2=good progress, 3=good solution.
Homework Assignments:
Graded problems in red
Ch. 1: 1,3,4,6,7,8,9,13,14,15;
Additional (required!) problem: Find all subspaces of R^2. Justify your answer (ie, explain why you know you've found ALL subspaces)
Ch. 2: 3,5,7,8,9,10,11,13,14,15,16;
Additional (required!) problem: Suppose that W_1 and W_2 are subspaces of V and have basis sets B_1 and B_2 respectively. Suppose also that the intersection of W_1 and W_2 contains only the zero vector. Prove that the union of B_1 and B_2 is a basis for the direct sum of W_1 and W_2.
Ch. 3: 1,2,3,4,5,7,8,9,10,11;
Additional (required!) problem: Suppose that V is the direct sum of subspaces U_1 and U_2. Show that for W any vector space, L(U_1, W) and L(U_2, W) are naturally subspaces of L(V, W) and it is a direct sum of them.
Ch. 3: 12,13,14,15,16,20,22,23,24,25,26;
Additional (required!) problem: When does a 2x2 matrix with real entries give an isomorphism from R^2 to itself? Be sure to justify your answer.
Ch. 4: 1,2,3,4,5;
Additional (required!) problems: 1) Let V be a vector space over the complex numbers. Show that V is naturally a vector space over the real numbers by forgetting the possibility of scaling by imaginary numbers. 2) Suppose v_1, ..., v_n is a basis for V as a vector space over the complex numbers. Show that v_1, iv_1, ..., v_n, iv_n is a basis for V over the real numbers. 3) Calculate the matrix of the linear transformation T:V--->V given by scaling by 1+i with respect to the basis v_1, iv_1, ..., v_n, iv_n.
Ch. 5: 1,2,3,4,5,6,7,8,9,10,11,12
Ch. 5: 13,14,15,16,17,18,19,20,21,22,23,24
Ch. 6: 1,2,3,4,5,6,7
Ch. 6: 9,10,11,12,13,14,15,16,17,18,19,20
Ch. 6: 21,22,26,27,28,29,30,31,32
Ch. 7: 2,3,4,5,6,7,8,9,10,11
Here are the worksheets that are handed out during discussion section. You should use these worksheets to get extra practice at computations. They will also highlight various consequences of Theorems you will see in this course. If you have any questions on the worksheets then please get in contact with me; better still, ask a question at piazza.com (making sure to remember to state which problem you are working on!)