MAT 240F - Problem Set 8
Not to be handed in.
Remark: With the exception of questions 5c) and 6d), the questions in this problem set can
be used as review questions for the nal exam.
1. A matrix A = (ajk ) Mnn (F ) is upper triangular if ajk = 0 whenever
MAT 240 - Problem Set 3
Due Thursday, October 9th
Questions 3a), 4a), 5b), 9c), 10 and 11a) will be marked.
1. Let x = (x1 , x2 , x3 , x4 ) C4 . Set y = (0, 1, 1 + i, i) and z = (1, i, 1, i). Show that
x span(cfw_y, z) if and only if x3 = (2 i)x1 + (1 + i
MAT 240 - Problem Set 4
Due Thursday October 16th
Questions 1d), 2c), 2d), and 7 will be marked.
1. In each case, nd a basis for the indicated subspace W of the vector space V . In some parts,
you must nd a basis for W that has certain additional properti
MAT 240 - Problem Set 6
Due Thursday, November 13th
Questions 3, 4a), and 5e) will be marked.
1. #2 b) 2.3.
2. #12 2.3.
3. Let V be a vector space over a eld F and let L(V ) be the vector space of linear transformations
from V to V . Suppose that T L(V ).
MAT 240 - Problem Set 1
Due Thursday, September 25th
NOTE:
Questions 4, 5, 6a), 9b) and 10c) will be marked. It is not necessary to hand in solutions to
other questions, but it is recommended that students work through at least two thirds of the
question
MAT 240 - Problem Set 5
Due Thursday November 6th
Questions 1a), 8, and 11a) will be marked.
1. Determine whether the function T : V W is a linear transformation.
a) Let V = W = P (F ), where F is a eld. Dene T (f )(x) = xf (x2 + 1) + f (1)(x3 1),
f V.
b)
MAT 240F - Problem Set 7
Due Thursday, November 20th
Questions 1, 3 b), 4 c) and 5 will be marked.
1. Let V1 , V2 , W1 and W2 be vector spaces over a eld F . Let T L(V1 , V2 ), U1 L(W1 , V1 ) and
U2 L(V2 , W2 ). Suppose that nullity(T ) is nite and U1 and
Do not turn this page until instructed.
Math 240 Algebra I Term Test
University of Toronto, October 25, 2012
Solve 4 out of the 5 problems on the other side of this page.
Each of the problems is worth 25 points.
You have an hour and 50 minutes.
Notes.
No
Dror Bar-Natan: Classes: 2014-15:
http:/drorbn.net/?title=14-240
Do not turn this page until instructed.
Math 240 Algebra I
Term Test
University of Toronto, October 21, 2014
Solve 4 of the 5 problems on the other side of this page.
Each problem is worth 2
MAT 240 - Problem Set 2
Due Thursday, October 2
Questions 1c), 1d), 1e), 3a), 4b), 4c), and 5 will be marked.
1. In each of the following parts, determine whether or not the set V with the given operations
of addition and scalar multiplication, is a vecto