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Math 232 , Spring 2007
Final Examination
April 10 , 200 7, 15:3 0 – 18 :
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 11 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. The last question is a bonus question. This question receives no partial credit.
9. All the best.
Bonus question
Question Points Score
Total: 60
- Consider the matrix
(a) (3 points) Compute det(A).
Answer
(b) (3 points) Compute A
− 1
.
Answer
(c) (3 points) Solve the system
2 x 1 +8x 2 − 2 x 3
3 x 1
+1 3 x 2
− 4 x 3
− 1 x 1
− 7 x 2
+3x 3
- Let W ⊂ R
3
be the subspace spanned by {
(a) (3 points) Compute a basis for W
⊥
. Explain your method.
Answer
(b) (3 points) Compute the orthogonal projection of
onto W and onto W
⊥
. Explain
your method.
- Consider the following basis for R
3
: {
(a) (2 points) Give the definition of an orthogonal basis.
Answer
(b) (2 points) Show that the given basis for R
3
is not an orthogonal basis.
- An experiment has yielded the following measurements for the quantities (x, y): ( 0 , 1), (1, 3), (2, 3), (3, 1).
Some theory predicts that y = β 0
(a) (2 points) Express the problem of determining β 0
, β 1
as a system of linear equations. Is
the system consistent? Explain why (not).
Answer
(b) (2 points) For a system of linear equations Ax = b, describe the least squares solution to
this system in terms of A, b and orthogonal projections.
Answer
(c) (4 points) Compute the least squares line for the points given.
- (3 points) Let A be an n × m matrix. Prove that if v ∈ Nul(A) then v is orthogonal to all
vectors in the row space of A.
- Let v ∈ R
n
and consider the map T : R
n
→ R defined by T (w) = w · v.
(a) (2 points) What properties should T satisfy to be a linear transformation?
Answer
(b) (3 points) Prove that T is a linear transformation.
Answer
(c) (3 points) Give the standard matrix of T and describe how you compute it.
Bonus question: Let W ⊂ R
n
be a subspace and let proj W
: R
n
→ R
n
be the orthogonal
projection onto W. Prove that proj W
can only have the eigenvalues 0 and 1 and describe the
corresponding eigenspaces in terms of W.
