Math 3130 Spring 2016 Abrams
Exam 3 Information
This exam will be given during the Final Exam period for this course, which is Thursday, May 12.
Although officially the final exam period starts at 12:40 that day, EXAM 3 WILL BE DISTRIBUTED IN
CLASS AT 1:20. You will have until 2:40 to finish Exam 3.
Reminder of the Calculator Policy: You may use (but will not need) a calculator which has no graphing
capability, no internet connectivity, and no QWERTY keyboard.
Exam 3 will cover the material in Chapter 6 (sections 1,2,3,6) and Chapter 5 (sections 1, 2, 4).
I would suggest you read your notes to get a good ’big picture’ of what we have covered. Then review your
homework. For problems you think you are having trouble with, do a few more similar problems in each of the
assigned sections. Only then should you attempt the Practice Exam. Complete the practice exam as if you were
in a real exam setting (closed notes, 75 minutes, etc ...) Compare your answers to those on the answer sheet.
The answers are posted on the course website
http://www.uccs.edu/∼gabrams/CourseMainPageMath3130Spring2016.html
For those problems you get right, you can probably assume that you know the information on that topic
well enough for the exam. For those you got wrong, go to some similar problems from the homework and try to
practice those. If you still need assistance, SEE ME, or use any of the other additional resources which support
this course. (Office hours, Supplemental Instruction, and Math Center hours are listed below.)
I will give you the definition of an inner product (p. 345) if necessary.
Be able to show that a given association of a real number with each pair of vectors in a vector space is or
is not an inner product.
Be familiar with examples of inner products on vector spaces (e.g. Euclidean inner product on Rn , definite
integral inner product on Pn , etc.)
Know the definition of the length of a vector in an inner product space (p. 346). Be able to compute
lengths for specific examples.
Know the definition of orthogonality of two vectors in an inner product space (p 357). Be able to show
that two given vectors in an inner product space are or are not orthogonal.
Be able to prove the Generalized Theorem of Pythagoras (p. 358).
Know the definition of an orthonormal basis for an inner product space.
Know how to use Theorem 6.3.2 to get the representation of a vector in terms of an orthonormal basis.
Be able to prove this theorem as well. I will give you the statement of Theorem 6.3.2a if needed; you will need to
know the statement of Theorem 6.3.2b.
I will give you the statement of Theorem 6.3.3. Given an orthonormal basis of a subspace W of V , be
able to compute the quantities w1 and w2 in this theorem. Be able to prove that w2 is in fact orthogonal to all
vectors in W .
I will give you the steps of the Gram Schmidt process (pp. 370-371). Know how to use these steps to
produce an orthonormal basis for a vector space.
Know the definition of an eigenvalue and eigenvector for a square matrix A.
Be able to prove that if λ is an eigenvalue for A, then det(A − λI) = 0. Conversely, be able to prove
that if det(A − λI) = 0, then λ is an eigenvalue for A.
(over for more info)
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Be able to compute the eigenvalues of a matrix A. Be able to compute the corresponding eigenspaces for
each of these eigenvalues. Describe these eigenspaces both parametrically, and by giving a basis.
Know the result of Theorem 5.2.1; you need not be able to prove it.
Be able to ’diagonalize’ a given square matrix A. Produce the matrix P which diagonalizes A. Write
down the matrix product P −1 AP without directly computing (i.e., know that this matrix product will yield the
eigenvalues along the main diagonal in the appropriate order). Interpret the matrix equation P −1 AP = D in
terms of the Linear Transformation Theorem of Chapter 8.
Be able to find a basis B for the vector space V so that the matrix of a given linear transformation
T : V → V with respect to this basis is diagonal.
Be able to compute (easily) the powers of a square matrix by the procedure described at the end of section
5.2.
Be able to solve a system of linear first order differential equations by using the diagonalization procedure
of section 5.4.
I’ll give you the orthonormal basis (5) of the
subspace Wn presented on p. 397 for the vector space C[0, 2π]
R 2π
with respect to the inner product hf (x), g(x)i = 0 f (x)g(x)dx. Using this, be able to write down each of the
Fourier coefficients a0 , a1 , ..., an , b1 , ..., bn .
ADMINISTRATIVE STUFF
There will be NO class session on Thursday May 5. ( I’ll have some office hours during that time instead, see
specifics below.)
My office hours between now and Exam 3: (all office hours are held in Engineering Building Room 288):
- Th May 5, 3:45 - 4:30 (note: this is during what normally would be class time)
- Tu May 10, 11:15 - 12:00
- W May 11, 3:00 - 3:30
- Th May 12, 12:45 - 1:15
and by appointment.
Supplemental Instruction Sessions with Ikko Saito:
- As usual during the week of May 2 - 6: Tuesday and Thursday, 12:20 - 1:50pm, Columbine Room 105
- During “Finals” week: Dates and times and locations TBD.
Mathematics Center: ENGR Room 233. Drop-in tutoring available Monday -Thursday 8 am - 7 pm; Friday
8 am - 4:30 pm; Sunday 11 am - 3 pm.
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