Math 217 Final Problems
This is an open book Final examination. After you start reading the problems you have 24 hours to work on them, but keep in mind that all solutions should be submitted by 9 am on May 18th. Attach this page to the solutions. Put you
MAT 217 Spring 2012
Homework 10
Due Mon. Apr. 23
Notation:
1. For a complex number z , we denote the complex conjugate by z , i.e. if z = a + ib, with
a, b R, then z = a ib.
2. If V is an F -vector space, recall the denition of V V : it is the F -vector s
MAT 217 Spring 2012
Homework 9
Due Mon. Apr. 16
Notation:
1. If F is a eld then we write F[X ] for the set of polynomials with coecients in F.
2. If P, Q F[X ] then we say that P divides Q and write P |Q if there exists S F[X ]
such that Q = SP .
3. If P
MAT 217 Spring 2012
Homework 8
Due Mon. Apr. 9
Problem 1:
Let V be an F-vector space and let W1 , . . . , Wk be subspaces of V . Recall the denition of
the sum k=1 Wi . It is the subspace of V given by
i
cfw_w1 + + wk : wi Wi .
Recall further that this su
MAT 217 Spring 2012
Homework 3
Due Mon. Feb. 27
Notation:
1. A group is a pair (G, ), where G is a set and : G G G is a function (usually
called product) such that
(a) there is an identity element e; that is, an element with the property
e g = g e = g for
MAT 217 Spring 2012
Homework 6
Due Mon. Mar. 26
Notation:
1. n = cfw_1, . . . , n is the nite set of natural numbers between 1 and n;
2. Sn is the set of all bijective maps n n;
3. For a sequence k1 , . . . , kt of distinct elements of n, we denote by (k1
MAT 217 Spring 2012
Homework 12
Due Wed. May 9
Notation:
1. Recall that if V is a vector space over the real numbers, we dene its complexication
by
VC = V V = cfw_(v1 , v2 ) : v1 , v2 V
and (complex) scalar multiplication is done as follows. For z = a +
Lecture 19
We have discussed determinants of matrices (and of n vectors in F n ). We will now dene
for transformations.
Denition 0.1. Let V be a nite dimensional vector space over a eld F . If T : V V is
linear, we dene det T as det[T ] for any basis of V
1. (a) Let A be an n n matrix (over C). The Cayley-Hamilton theorem says that A
satises its own characteristic polynomial; that is, if pA (x) is the characteristic
polynomial of A, then we have pA (A) = 0. Suppose that A is invertible. Then
use the Cayley
MAT 217 Spring 2012
Homework 11
Due Mon. Apr. 30
Notation:
1. For all problems below, F is a eld of characteristic dierent from 2, and V is a nitedimensional F -vector space. We write Bil(V, F ) for the vector space of bilinear forms
on V , and Sym(V, F )
MAT 217 Spring 2012
Homework 2
Due Mon. Feb. 20
Notation:
1. If F is a eld then for m, n N we write Mmn (F ) for the vector space of mn matrices
with entries from F . Addition of matrices is done component-wise: for matrices A and
B , the (i, j )-th entry