## Math 2230, Linear algebra and multivariable calculus

The final exam (with answers) grade breakdown: D < 15 < C < 30 < B < 50 < A.

Book: Hubbard and Hubbard Vector calculus, linear algebra and differential forms 4th edition

Scroll down for info on the final. As with the midterm, you can bring in one double-sided sheet of handwritten notes. (Written by you!)

The midterm was Thursday Oct 9, in class. Here it is, with answers.

Office hours during the term (in my office, 515 Malott):

• Monday 1-2 PM
• Tuesday after class
• Monday 12-1 is also possible, but you must email me by Sunday evening to say you want to drop by then.
• Homeworks:
• Homework #1 was due Sep 4
• Homework #2 was due Sep 11
• Homework #3 due Sep 18
• Homework #4 due Sep 25: 1.5.4abc, 1.5.6, 1.5.10, 1.5.20
• Homework #5 due Oct 2: 1.6.2, 1.7.5, 1.7.8, 1.7.13, 1.7.19
• Good questions to prepare for the midterm:
• 1.1.4, 5, 9
• 1.2.2, 4, 5, 6, 8, 10, 12, 13, 16, 20, 23
• 1.3.2, 4, 7, 9 & 16, 10, 11, 19
• 1.4.1, 6, 7, 13, 24, 26
• 1.5.1, 3, 4, 5, 10, 19, 20, 23
• 1.6.2, 3, 11
• 1.7.4, 10, 13, 15, 19, 20
• though the test may be a little more proofy than these questions.
• HW #6, due 10/23 is at this link. NOTE I added a line to question #3 to help avoid confusion in a definition.
• HW #7, due 10/30:
• 2.2.2, 2.2.5, 2.3.3, 2.3.6, 2.4.5, 2.4.13
• Let M have echelon form F, and its transpose M^T have echelon form G. Use bignesses to show that F,G have the same number of pivots.
• Let S be a subset of a vector space V. Give a necessary and sufficient condition, in four words or less, for S = span(S). Prove that's necessary, and prove that it's sufficient.
• HW #8, due 11/6 (sorry for the Hallowe'en-related delay):
• 1. We proved that if B_1, B_2 are bases, and b is a vector we don't like having in B_1, we can remove it and put in some element of B_2 to get a new basis.
• What if we only said B_1 spans, and we want to get a new spanning set? Is it still true? If yes prove it, if not give a counterexample.
• Or what if we only said B_1 is independent, and we want to get a new independent set? Is it still true? If yes prove it, if not give a counterexample.
• For whichever of those above are true, what if we then weaken "B_2 is a basis" to only be spanning? If that's still good enough, prove it; if not, give a counterexample.
• Same question except that B_2's condition is weakened to being independent.
• 2.4.12, 2.5.2, 2.5.4, 2.5.8, where ker(f) := {v : f(v) = 0}, 2.5.15
• HW #9, due 11/13: 2.6.3, 2.6.4 (hint: pick and extend bases), 2.6.7, 2.6.11, 2.7.1, 2.7.3, 2.7.4
• HW #10, due 11/20:
• Modify the proof from class, that every matrix is upper-triangularizable, to show that one shows we can upper-triangularize in such a way that all the diagonal entries equal to a given number come in succession.
• If S is a set of vectors that is linearly independent and orthogonal, show that S can be extended to a basis with all vectors orthogonal. (Add one vector at a time, and if it's not orthogonal to the previous vectors, fix it.)
• If M is symmetric, and v is in ker(M^k) for some power k, show v in ker(M). Give a counterexample for nonsymmetric M.
• 2.7.6, 2.10.1, 2.10.5, 2.10.10
• HW #10, due 12/4 (last day of class):
• 3.1.2, 3.1.3, 3.1.7, 3.1.10, 3.1.19
• 3.2.1, 3.2.6, 3.2.7
• 3.3.9

• Final exam: Mon, Dec 15 9:00 AM in Malott 406.

Book sections covered on the final:

• Ch 0
• Ch 1.0-1.8
• Ch 2.0-2.7, 2.10
• Ch 3.0-3.2, 3.5-3.6
• Some review exercises:
• The ones above for the midterm
• 2.1.2, 5
• 2.2.4, 9
• 2.3.2, 13
• 2.4.3, 4, 5 (i.e., show the span is contained in any other such subspace), 8
• 2.5.3, 6, 9, 17, 21
• 2.6.9
• 2.7.2
• 2.10.2, 4, 8
• 2.11: 2.3, 5, 8, 11, 16, 29
• 3.1.5, 12*, 22
• 3.2.5, 3.2.11e, 3.2.12
• Also, you should be clear on the two formulations of the spectral theorem we proved in the last week: If M is a real symmetric matrix, then
• (v1) there exists an orthonormal basis of real eigenvectors
• (v2) there exists an orthogonal matrix B with det=1 such that B^T M B is diagonal.
• I'll have office hours Tuesday @1 PM, Thursday & Friday @10 AM, in 515 Malott. Nothing structured; come with questions about class or (particularly Friday) about the review questions above. Amin will have office hours @4:30 PM Friday in the normal office hours room.