LECTURE 2
Dention. A subset W of a vector space V is a subspace if
(1) W is non-empty
(2) For every v , w W and a, b F, av + bw W .
Expressions like av + bw, or more generally
k
ai v + i
i=1
are called linear combinations. So a non-empty subset of V is a
LECTURE 21: SYMMETRIC PRODUCTS AND ALGEBRAS
Symmetric Products
The last few lectures have focused on alternating multilinear functions. This one
will focus on symmetric multilinear functions. Recall that a multilinear function
f : U m V is symmetric if
f
LECTURE 23: METRIC SPACES
This lecture will look at the topological properties that arise from the notion of
distance provided by the norm.
Denition 1. If V is an inner product space, then the distance between and
v
u
is
d( , ) = | |.
vu
vu
Proposition 1
PROBLEM SET #1
DUE SEPTEMBER 8
1. Find 2 distinct bases for Q2 .
2. Find all 3 bases of F2 .
2
3. Are the vectors (1, 1, 1), (0, 1, 2), (2, 1, 3), and (4, 0, 11) linearly independent?
If not, nd a dependence relation.
4. Let a, b F and let v, w V . Show t
LECTURE 28: ADJOINTS AND NORMAL OPERATORS
Todays lecture will tie linear operators into our study of Hilbert spaces and
discuss an important family of linear operators.
Adjoints
We start with a generalization of the Riesz representation theorem.
Theorem 1
LECTURE 24: ORTHOGONALITY AND ISOMETRIES
Orthogonality
Denition 1. If v and u are vectors in an inner product space V , then u and v
are orthogonal, written u v , if
u, v = 0.
Since V is an inner product space is orthogonal to is a symmetric relationship.
PROBLEM SET #2
DUE OCTOBER 6
1. Basic Properties of the Determinant. In this problem, let A be an n n
matrix over F.
(1) Using the denition, show that if A has two rows which are scalar multiples
of each other (so for some i and k , ai,j = ak,j for all j
Chapter 69 NEIGHBORHOOD ENTERPRISE ZONES
Sec. 5.551. Purpose.
Sec. 5.552. Definitions.
Sec. 5.553. Designation of Zones.
Sec. 5.554. Application for Certificate.
Sec. 5.555. Inspections.
Sec. 5.556. Issuance of Certificate of Compliance.
Sec. 5.557. Inspe
import java.util.*;
import java.text.*;
public class FutureValueApp
cfw_
public static void main(String[] args)
cfw_
/ display a welcome message
System.out.println("Welcome to the Future Value Calculator");
System.out.println();
/ perform 1 or more
import java.util.Arrays;
import java.util.Comparator;
import java.util.Scanner;
public class SortedCustomersApp
cfw_
public static void main(String[] args)
cfw_
System.out.println("Welcome to Soreted Customers App\n");
final int NUM = 5; /Setting Array Si
import java.util.Scanner;
public class TestScoreApp cfw_
public static void main(String[] args) cfw_
Scanner sc = new Scanner(System.in);
String choice = "y";
do cfw_
int scoreTotal = 0;
int scoreCount = 0;
int testScore = 0;
int numScores;
System.out.pri
import java.util.Scanner;
public class TestScoreApp cfw_
public static void main(String[] args) cfw_
/ display operational messages
System.out.println("This Java application has run
successfully.");System.out.println("This Java application has tun
success
import kareltherobot.*;
public class SimpleApplication implements Directions
cfw_
public static void main(String [] args)
cfw_ UrRobot karel = new UrRobot(2, 2, East, 0);
/harvest first row of beepers
karel.move();
karel.pickBeeper();
karel.move();
karel.
/*ASH Implementing solutions for Exercise 7-2
*Took the get methods and put the in the Validator
*Class. Made the WithinRange methods override the
*regular methods. Call the validator class in the main app.
*Made another class called FinancialCalculations
public class FinancialCalculations
cfw_
public static double calculateFutureValue(double monthlyInvestment, double
monthlyInterestRate, int months)
cfw_
double futureValue = 0;
for (int i = 1; i <= months; i+)
cfw_
futureValue =
(futureValue + monthlyInve
LECTURE 20: APPLICATIONS OF EXTERIOR PRODUCTS
Today will be a little dierent. Rather than focus too much on theory, we will
look at two very interesting examples involving the exterior product. But before
then, we have to talk a little more about functori
LECTURE 22: INNER PRODUCTS
For the remaining lectures, we will work exclusively over R or C. Well also look
at a very special kind of bilinear form.
Denition 1. An inner product is a function , : V V F such that
vv
v 0.
(1) , 0 with equality i =
cfw_
,
LECTURE 3
(TYPED!)
Dention. A basis is a linearly independent spanning set.
Theorem. Every vector space has a basis.
We wont prove this; its actually essentially equivalent in the innite dimensional
case to one of the axioms of set theory: the axiom of ch
LECTURE 8 - DUALS CONTINUED
We saw last time that to any vector space V , we can naturally associate another
vector space, the dual space V of linear functional V F. This had the property
that if L : V W , then there is a linear map L : W V dened simply b
LECTURE 9 - EIGENVALUES
Having spent some time looking at the general properties of vector spaces and
the generic interplay with linear transformations, we turn our attention now to best
understanding a given linear operator on a given vector space. Thus
LECTURE 16: TENSOR PRODUCTS
We address an aesthetic concern raised by bilinear forms: the source of a bilinear
function is not a vector space. When doing linear algebra, we want to be able to
bring all of that machinery to bear. That means we need to cons
LECTURE : BILINEAR FORMS
Today we will focus on extra structure we can put on a vector space that render
certain constructions more natural.
Denition. A bilinear form , on V is a function
, : V V F
that is bilinear:
av + bu, w = a v , w + b u, w and
v , a