Assignment 1, due at classtime, Friday, September 28
1. Let n be a positive integer. We have dened addition, which for this problem we will
denote by , and multiplication, which for this problem we will denote by , on Zn =
cfw_ 0, 1, 2, . . . , n 1 by th
Assignment 6, due at classtime, Friday, November 2
1. Let V and W be vector spaces over a eld F . Then F (V, W ), the set of all functions from V to
W , is a vector space over F , with addition and scalar multiplication as dened in class using the
additio
Assignment 4, due at classtime, Friday, October 19
1. Let u1 = (2, 3, 1), u2 = (1, 4, 2), u3 = (8, 12, 4), u4 = (1, 37, 17), and u5 =
(3, 5, 8). You may assume that S = cfw_ u1, u2 , u3 , u4 , u5 spans R3 . Find a subset B of
S that is a basis for R3 .
S
Assignment 2, due at classtime, Friday, October 5
1. Let F be a eld, S = cfw_ 1, 2, 3 , and consider the F -vector space F (S, V ). For each
0 if i = j
for j S . Prove that
i = 1, 2, 3, let ei : S F be dened by ei (j ) =
1 if i = j .
cfw_ e1 , e2 , e3 is
Assignment 3, due at classtime, Friday, October 12
1. Let V be a nite dimensional vector space over C. Since R is a subeld of C, we may consider V
as a real vector space by restricting the scalar multiplication to R (that is, scalar multiplication
is a fu
Assignment 7, due at classtime, Friday, November 9
1. Let U , V , W be vector spaces over a eld F , with U nite dimensional. Let T : V W be
a linear transformation. Dene a function LT : L (U, V ) L (U, W ) by LT (R) = T R for all
R L (U, V ). In Mondays c
Assignment 8, due at classtime, Friday, November 16
1. Let T be the linear transformation from M22 (Q) P2 (Q) given by T (E1,1 ) = x, T (E1,2 ) = x +1,
T (E2,1 ) = x2 + x, and T (E2,2 ) = 1 + x + x2 .
(a) For any a, b, c, d Q, nd T (
Solution: We have T (
Mathematics 2120A Final Exam
December 10, 2012
5
marks
9:00am-noon
1. Let F be a eld. In the last assignment, you proved that for positive integers m and n, if A Mmm (F ),
B Mnn (F ), and C Mnm , then det(
A|O
C|B
= det(A) det(B ). You may assume this r
Assignment 11, due at classtime, Wednesday, December 5
1. (a) Prove by induction on n that if A is a square matrix with entries from a eld F , and A has
the form A =
B|O
C|D
where B is an n n matrix and D is a square matrix, then
det(A) = det(B )det(D )
Assignment 9, due at classtime, Friday, November 23
1. Let A M34 (R) be given by A =
Ir | O
O|O
P AQ =
213 4
332 1
4 5 1 2
. Find invertible matrices P and Q such that
for some positive integer r .
Solution: Note that we can keep track of the succession o
Assignment 10, due at classtime, Friday, November 30
1. Let F be a eld, and let A Mmn (F ). Prove that there exist B1 , B2 , . . . , Bn Mmn (F ) such
that A = B1 + B2 + + Bn and for each i, rank (Bi ) 1.
Solution: For each i = 1, 2, . . . , n, let Bi deno