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Math 232, Fall 2007
Final
Dec. 12, 2007
Last Name:
First Name:
SFU ID:
1. DO NOT LIFT UP THE COVER PAGE UNTIL INSTRUCTED.
2. No calculators are allowed.
3. This test is comprised of 18 pages (including cover page)
4. Once the test begins, please check that all pages are intact.
5. Do ALL questions.
6. Clearly explain your answer. No credit will be given for just writing down the
answer.
7. If the answer space provided is not sufficient, write your answer on the back
of the previous page. Clearly mark the question number.
8. Good luck.
Question Points Score
Total: 100
- (9 points) Assuming that all the flows are nonnegative in the figure below, what is the largest
possible value for x 3? Show all work.
x
x
x
1 x
- Which of the following sets are subspaces of R^4? Justify your answer. (a) (3 points)
W 1 =
x y z w
:^ x^ + 3y^ = 1 and^ x^ + 2z^ = 3
(b) (3 points)
W 2 =
x y z w
:^ x^ + 3y^ = 0 and^ x^ + 2z^ = 0
(c) (3 points)
W 3 =
x y z w
:^ x^ + 3y^ = 0 or^ x^ + 2z^ = 0
- In the table below, data are given for two variables x and y.
x y -2 - -1 - 0 1 1 3 2 4
(a) (7 points) Compute the line of best fit for these data. Show all work. (b) (2 points) What does the line of best fit estimate the value of y will be when x = 5.
- Suppose T : R^2 → R^2 is given by
T
([ (^) x 1 x 2
])
[ (^2) x 2 + 3x 1 x 1 + x 2
]
and B =
{[ 1
]
[ 1
]}
and C =
{[ 0
]
[ 1
]}
(a) (7 points) Compute the matrix of T relative to the bases B and C, [T ]CB. Show all work. (b) (2 points) if [~x]B =
[ 1
]
what is [T (~x)]C?
- (a) (8 points) Find the orthogonal projection of
~v =
onto the Column space of the matrix
A =
Show all work. (b) (2 points) Find ~y ∈ Col(A) and ~z ∈ Col(A)⊥^ such that ~v = ~y + ~z.
- Let A =
(a) (6 points) Find an invertible matrix S and a diagonal matrix D such that A = SDS−^1. Show all work. (b) (5 points) Find a formula for A^100. (Hint: Use part a. Your formula should be given by a 2 × 2 matrix whose entries are of the form C 0 a^100 + C 1 b^100 , where C 0 , C 1 , a, b are constants.) Show all work.
