## Math 2240, Linear algebra and multivariable calculus

Final exam: Tuesday, May 12 2:00 PM in Malott 406.

Sections we focused on on the final:

• 3.7
• 4.1,3,4,5,8,10,11
• 5.2
• 6.1,2,3,4,6,7,9
• Book: Hubbard and Hubbard Vector calculus, linear algebra and differential forms 4th edition

Office hours: Monday 12-2, Tuesday after class.

Homework #1 (due Thursday 2/5/15):
• 3.7 #1,4,11b,13,17
• 4.1 #9,11,15
• Homework #2 (due Thursday 2/12/15):
• 4.3 #1,3,5
• 4.4 #2 (balls vs. boxes), 3, 5
Turns out #3 has been struck from the second printing of the book! So instead I'm putting in this question in its place:
• Find a subset S of [0,1] that is not pavable, and you can prove does not have measure zero.
• Two questions: if A has measure (resp. volume) zero, and B is a subset of A, then B has measure (resp. volume) zero.
• Very sneaky definitional question: find a set of reals with measure zero, but not volume zero.
• Homework #3 (due Thursday 2/19/15):
• 4.5 #2,3,5,7,8,15
• Homework #3 (due Thursday 2/26/15):
• 4.8 #2,10,19
• 4.9 #3,4
• Homework #4 (due Thursday 3/5/15):
• 4.10 #3,5,7,12,13,19
• Homework #5 (due Thursday 3/12/15 3/19/15):
• 4.11 #1, 8, 11, 13, 18

• Midterm exam: March 17 in class. Test with answers here. Grade distribution: C < 25 < B < 50 < A.
People did really well, I'm proud of you!!

Homework #6 (due Thursday 3/26/15):

• Let 0 -> A_1 -> A_2 -> ... -> A_d -> 0 be a series of linear maps f_1...f_{d-1}, where each ker(f_i) = image(f_{i-1}). Prove that the alternating sum of the dimensions of the A_i is zero.
• Simultaneously construct bases for the A_i, such that f_i(a basis element of A_i) is either a basis element of A_{i+1}, or zero.
• Let A < B < C be three finite-dimensional vector spaces. Find the natural map C/A -> C/B and compute its kernel.
• Let A,B < C be two subspaces, and consider the map C -> C/A x C/B taking c |-> (c + A, c + B). What's its kernel?
• The number of subsets of an n-element set is 2^n. Figure out the number of quotients for small n, e.g. q(1)=1, q(2)=2. Use that experimental data to locate the sequence at The On-Line Encyclopedia of Integer Sequences. In particular, what sequence number is it, there?
• (The last question was inserted as #2; look up there.)
• Homework #7 (due Thursday 4/9/15):
• Define V @ W := Hom(V*,W). Find a natural correspondence s: V @ W -> W @ V, when W,V are finite-dimensional.
• Restrict your correspondence to s: V @ V -> V @ V. If dim V = n, then s is a map from an n^2-dimensional space to itself. What are its eigenvalues? What are their multiplicities?
• We defined "dx_{i_1} ^ ... ^ dx_{i_r}" in class, when i_1 < ... < i_r, and we also defined "^" separately. Let I,J be two subsets of {1,...,n} of sizes r,m, and let alpha = "dx_{i_1} ^ ... ^ dx_{i_r}" ^ "dx_{j_1} ^ ... ^ dx_{j_m}".
• If I,J are not disjoint, show alpha = 0.
• If I,J are disjoint, and K is their union, show alpha is a multiple of "dx_{k_1} ^ ... ^ dx_{k_{r+m}}".
• Homework #8 (due Thursday 4/16/15):
• 6.1 #6,10
• 6.2 #1,3
• 6.3 #2,6,10,14
• Homework #9 (due Thursday 4/23/15):
• Let 0 -> V_1 -> V_2 -> ... -> V_n -> 0 be an exact sequence (of finite-dimensional vector spaces), i.e. the kernel of each map equals the image of the previous. Show that, given orientations on all but one of these spaces, one can put a natural orientation on the missing space.
• Homework #10 (due Thursday 4/30/15):
• 6.4 #1,4,9
• 6.7 #1,2,9
• Homework #11 (due Wednesday 5/6/15, in section):
• 6.6 #5
• 6.9 #1,4