MA 237 Fall 2008
Lecture #2
August 20
BRING TO CLASS:
This lesson plan and Penney book.
Folders, syllabi and queries for ~ new 5 students.
Racetrack game (40)
Racetrack Overhead
Overhead Pen
TODAY WE WILL:
Take Attendance
Lecture on Vector Spaces
I.
Spatial Vectors (RACETRACK GAME)
II.
Matrices
III.
Definition of a Vector Space
Assign homework and quiz topic.
Lecture: Vector Spaces
I. Spatial Vectors
From Physics: A spatial vector is a geometric object that has magnitude and
direction.
In 2D, an ordered pair (x,y) of 2
In 3D, an ordered pair (x,y,z) of 3
The ordered pair (x,y) can be interpreted in two ways:
(1) as the point (x,y)
(2) as the vector from (0,0) to (x,y)
Which one depends upon the context…
A scalar is a 1-component quantity that only has magnitude (e.g. a positive or
negative number  )
Spatial Vector Operations
Sum: (x1,y1)+(x2,y2)=(x1+x2,y1+y2)
Difference: (x1,y1)-(x2,y2)=(x1-x2,y1-y2)
Scalar Multiplication: c(x,y)=(cx,cy)
Example: Vector addition geometrically. Show that (2,1)+(2,1) = (4,2).
Example: Recall one form of a line:
{a+t*b | t  }
a and b are vectors
t is a scalar
 “belongs to”
 = set of real numbers
RACETRACK GAME (~ 10 Minutes)
II. Matrices
Def: An mxn matrix is an mxn array of numbers.
Is a matrix. Of what size?
 7 1 1
Example: Q = 
.
2x3 or 3x2?

 2 8 8
First # rows, then # columns!
Generally, a 2x3
matrix is
 a11 a12 a13 
A= 

a 21 a 22 a 23
The rows of A are a11 a12
 a11
The columns of A are 
,
 a 21
Where the (i,j)th entry of A is aij.
First index: row
Second index: column
a13 and a21 a22 a23.
 a12   a13 
a 22 ,  a 23 .

 

Notation: “A=[aij]” is shorthand for a matrix A with entries aij in each (i,j)th position.
Special Matrices
Example
mxn zero matrix: all entries are 0
0
0

1
0

0
0
0
nxn identity matrix: All entries are 0 except
aii=1 for all i=1,…, n
1
7
Transpose matrix: For A=[aij], AT is the
QT= 
matrix [bij] where bij=aij.
1
2
8
1 8
Matrix Operations
Sum: If A and B are matrices of the same size then the (i,j)th entry of A+B is aij+bij.
1 0 1 0 2 0
Example: 



1 0 0 1 1 1
Difference: If A and B are matrices of the same size then the (i,j)th entry of A-B is aij-bij.
1 0 1 0 0 0 
Example: 



1 0 0 1 1  1
Scalar multiplication: If c is a number and A is an mxn matrix, then the (I,j)th entry of
c*A is c*aij.
1 0 c 0
Example: c * 


1 0 c 0
What are matrices though exactly? We can add them, subtract them and multiply them by
scalars just like vectors. This motivates defining a new category “vector spaces” for
holding sets of objects that are like vectors in ‘important’ ways.
III. Vector Spaces
Let V be a set on which there are two operations: addition (+) and scalar multiplication
(*) so that A+B and c*A are defined in some way for A, B  V and a scalar c).
V is a vector space and the elements of V are vectors if:
A+B  V
 A,B  V
c*A  V
 A  V,  scalars c
A+B=B+A
 A,B  V commutativity
A+(B+C)=(A+B)+C
 A,B,C  V associativity

 A  V Zero vector
 0  V |A+0=A

 A  V Negative
 -A V |A+-A= 0
k*l*A=(k*l)*A
 A  V,  scalars k,l
k*(A+B)=k*A+k*B
 A,B  V,  scalars k
(k+l)*A=k*A+l*A
 A  V,  scalars k,l
1*A=A
 A  V multiplicative identity
CLOSURE
4 Rules for Vector
Addition
4 Rules for Scalar
Multiplication
Vector space examples: Spatial vecators, Matrices mxn, set of real polynomials.
The operations + and * can be strange but you would always be told how it is done.
Scalars can be elements of  or C (yielding complex or real vector spaces).
Example: Show that spatial vectors are commutative by showing that
(x1,y1)+(x2,y2)=(x2,y2)+(x1,y1). Generally: (x1+x2,y1+y2)+(x2+x1,y2+y1)
Show with (2,1) and (1,3).
Homework: page 15 #1a, #2 (so far)
Quiz Friday: page 15, #2 Commutativity and associativity of matrices by showing
A+B=B+A and A+(B+C)=(A+B)+C for given matrices A, B, C.
1
Example: A= 
2
Commutativity
2
A+B=B+A= 
5
3
1 2
 4 2
B= 
C= 



4
3 4
3 1 
Associativity
5
5 4 2 5
6 7 
A+(B+C)=(A+B)+C=A+ 
=
+C= 




8
6 5 5 8
8 9