ASSIGNMENT 2 - MATH 251, WINTER 2008 SOLUTIONS
1. Let B = cfw_(1, 1), (4, 5) and C = cfw_(2, 1), (1, 1) be bases of R2 . Find the change of basis
8
matrices B MC and C MB between the bases B and C. Le
ASSIGNMENT 7 - MATH 251, WINTER 2008 SOLUTIONS
(1) Use Proposition 7.2.9 to deduce Proposition 7.2.6 in the notes.
Proof. Consider the situation of a subspace U V . Let T be inclusion map
T : U V,
T (
ASSIGNMENT 8 - MATH 251, WINTER 2008 SOLUTIONS
(1) Let T : V V be a linear transformation. Let W be a subspace of V such that T (W ) W (such
a subspace is called a T -invariant subspace).
(a) Prove th
ASSIGNMENT 4 - MATH 251, WINTER 2008 SOLUTIONS
1. Deduce from the theorems on determinants the following:
(1)
(2)
(3)
(4)
If a column is zero, the determinant is zero.
det(A) = det(At ), where At is t
ASSIGNMENT 3 - MATH 251, WINTER 2008 SOLUTIONS
(1) Read 4.5 in the notes and write a proof for Proposition 4.5.2.
The proposition says:
Proposition 0.1. The following are equivalent:
(a) V is the inne
ASSIGNMENT 5 - MATH 251, WINTER 2008 SOLUTIONS
1. (A) Let W be a k-dimensional subspace of Fn . Prove that there are n k linear equations such that
W is the solutions to that homogenous system.
(B) Le
ASSIGNMENT 9 - MATH 251, WINTER 2008 SOLUTIONS
(1) Let A be a matrix in block form:
A1
0
A=
0
0
A2
0
.
0
.
.
Ak
Prove that
A = A 1 A 2 A k ,
and
mA = lcmcfw_mA1 , mA2 , , mAk .
You may use the formula
February 21, 2008
Quiz 2, MATH 251, Winter 2008
Time: 16:00 - 17:30.
PART I (35% of grade): multiple choice questions. Answer in the exam book. Choose one answer for each
question (there is only one c
March 20, 2008
Quiz 3, MATH 251, Winter 2008
Time: 16:30 - 18:00.
Answer the following questions. Write clearly and precisely, citing accurately results you
are using. Explain your calculations!
(1) L
January 31, 2008
Quiz 1, MATH 251, Winter 2008
Time: 16:00 - 17:30.
PART I (30% of grade): multiple choice questions. Answer in the exam book. Choose one answer for each
question (there is only one co
ASSIGNMENT 6 - MATH 251, WINTER 2008 SOLUTIONS
(1) Let A = (aij ) be a matrix in REF with the special columns being i1 , . . . , ir (so that A looks like
0 .
0 a1i1 . . . 0
0
.
0.
0 . . . . . . . . .
ASSIGNMENT 1 - MATH 251, WINTER 2008 SOLUTIONS
(1) The following are vector spaces (verify that to yourself). Determine in each case if they are nite
dimensional or innite dimensional by either provid
ASSIGNMENT 10 - MATH 251, WINTER 2007 SOLUTIONS
(1) Let A be the adjacency matrix of a k-regular graph G.
(a) Prove that k is an eigenvalue of A and nd the eigenvector.
(b) Prove that every eigenvalue