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Simon Fraser University
Math 232
Final Exam Date: 11 August 2007
Instructor : Aaron Bradford Time: 8:30 – 11:
Last Name (print): ________________________ First Name: _______________________
Signature: ______________________________ SFU Email ID: _____________________
Instructions:
1. DO NOT OPEN THIS EXAM UNTIL INSTRUCTED TO DO SO.
- Ensure that you have 13 pages of questions.
- No calculators, notes or books are allowed.
- Except for question 1, credit will not be given for answers with no explanation.
- Answer each question in the space provided. Continue on the back of the previous page if necessary.
- Have your picture ID ready for inspection.
- You will have 180 minutes to complete the exam. If you are caught writing after this time limit, you
will be assessed a 5 point penalty.
- With 10 minutes remaining in the exam, you will be asked to stay seated until time is up.
- Good luck!
Question 1 2 3 4 5 6 7 8 9 10 11 12 13 Total
Mark
Maximum 6 4 3 4 4 5 7 7 5 9 4 6 4 68
- ( ½ point each ) Mark the following statements as either true or false. No explanation is required.
a. ____ Whenever a system has free variables, the solution set contains many solutions.
b. ____ If a set contains fewer vectors than there are entries in the vectors, then the set is linearly
independent.
c. ____ A linear transformation :
n m
T → is completely determined by its effect on the
columns of the n × n identity matrix.
d. ____ A vector space is also a subspace.
e. ____ A linearly independent set in a subspace H is a basis for H.
f. ____ Row operations preserve the linear dependence relations among the rows of A.
g. ____ (^) If λ + 5 is a factor of the characteristic polynomial of A , then 5 is an eigen-value of A.
h. ____ If A is invertible, then A is diagonalisable.
i. ____ If A is diagonalisable and B is similar to A , then B is also diagonalisable.
j. ____ Every orthogonal set is linearly independent.
k. ____ If W is a subspace of
n
and if v
is in both W and W
⊥
, then v
must be the zero vector.
l. ____ (^) If A = QR is a QR-Factorisation of A , then the columns of Q form an orthonormal basis
for Col A.
- ( 1 point – 3 points ) The following system of linear equations has a unique solution:
1 2 3
1 3
1 2 3
x x x
x x
x x x
^ +^ +^ =
a. Cramer’s Rule gives a formula for i
x , i
th
entry in the solution vector x
. What is this formula?
b. Use Cramer’s Rule to calculate the value of 3
x.
- ( 4 points ) Let 2
P be the inner product space whose inner product is defined as
p t ( ) , q t ( ) = p ( − 1 ) q ( − 1 ) + p ( 0 ) q ( 0 ) + p ( 1 ) q ( 1 )
for any polynomials p and q in 2
P. Find an orthogonal basis for 2
P.
- ( 3 points – 3 points – 1 point )
^
B and
^
C are
two bases for a subspace W of 2 2
M
×
, the vector space of 2 × 2 matrices.
a. If
A W
then find [ A ]
B
, the B -co-ordinate vector of A.
b. Find P C ←B
, the change of co-ordinates matrix from the basis B to the basis C.
c. Calculate [ A ]
C
, the C -co-ordinate vector of A.
- ( 2 points – 3 points – 2 points ) Let
A
and
b
a. Give the matrix form of the normal equations.
b. Find all least-squares solutions of Ax = b
c. Compute the least-squares error associated with the least-squares solutions found in (b).
- ( 1 point – 1 point – 5 points – 2 points )
a. Define what it means for a matrix A to be orthogonally diagonalisable.
b. Let
A
. Without calculation, how do we know that A is orthogonally diagonalisable?
c. Find an orthogonal diagonalisation of A.
d. Using your work from (c), find a spectral decomposition of A.
- ( 2 points – 4 points )
a. Define what it means for a set W to be a subspace of V.
b. Let W be a subspace of
n
. Show that the set W
⊥
, the orthogonal complement of W , is also a
subspace of
n
.
- ( 1 point – 3 points )
a. Define what we mean when we say that two n × n matrices, A and B , are similar.
b. Show that if two n × n matrices are similar, then they share the same characteristic polynomial.
