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. Let v = ( 28 ) with respect to the standard
basis. Find [
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 (u) = u.
The map T : V U is just the restriction map, si
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 that there is a well dened linear map T induced by T , T
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 the transposed matrix.
If a row is zero, the determinant
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 inner direct sum of the subspaces U1 , . . . , Un ;
(b) V =
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) Let W1 = Span (cfw_(1, 0, 0, 1), (0, 1, 1, 1)) and W2 = S
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
Ab
1
0
Ab =
0
0
Ab
2
0
0
.
.
Ab
k
for every positive
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 correct answer). Do NOT write any explanations (they wil
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) Let W be the subspace of R4 dened by the equations
x1 +
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 correct answer). Do NOT write any explanations (they will
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 providing an innite independent set, or nding a
nite basis.
(
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 of A is real and satises | k.
(c) Prove that if G is n