© John Riley
2 September 2009
APPENDIX B: MAPPINGS OF VECTORS
B.1 VECTORS AND SETS
1
Orthogonal vectors and hyper-planes
Convex sets
Open and Closed Sets
B.2 FUNCTIONS OF VECTORS
8
Functions of 2 variables
Partial and total derivatives
Functions of n variables
Contour Sets
Concave and quasi-concave functions
Exercises
B.3
TRANSFORMATIONS OF VECTORS
29
Matrix
Quadratic Form
Quadratic Approximation of a function
Inverse matrix
Cramer’s Rule
Exercises
B.4
SYSTEMS OF LINEAR DIFFERENCE EQUATIONS
43
54 pages

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
Appendix B page
1
B
.1 VECTORS AND SETS
Key ideas: orthogonal vector, hyper-planes, convex sets, open and closed sets
We now extend our analysis to ordered
n
-tuples or “vectors.” Each component of
the vector
1
(
,...,
)
n
x
x
x
=
is a real number.
Where it is important to make clear the
dimension of the vector, we write
n
x
∈
\
. If each component is positive, then
x
is
positive and we write
n
x
+
∈
\
.
We will describe the vector
y
as being larger than the vector
x
if every component
of
y
is at least as large as
x
and write
y
x
≥
. If at least one component of
y
is strictly
larger we will say that
y
is strictly larger than
x
and write
y
x
>
. (We will occasionally
use the notation
x
y
±
to indicate that all components are strictly larger.)
A neighborhood of a vector is the set of points near it.
Thus to define a
neighborhood we need some measure of distance
( , )
d y z
between any two vectors
1
(
,...,
)
n
y
y
y
=
and
1
(
,...,
)
n
z
z
z
=
. If
1
n
=
, a natural measure is the absolute value of the
difference so that
( , )
d y z
y
z
=
−
.
Similarly, with
2
n
=
, a natural measure is the
Euclidean distance
y
z
−
between the two vectors.
Appealing to Pythagoras Theorem,
2
2
2
1
1
2
2
(
)
(
)
y
z
y
z
y
z
−
=
−
+
−
.
Extending this to
n
-dimensions, the square of Euclidean distance between ordered
n
-tuples is defined as follows.
2
2
y
z
−
1
1
y
z
−
y
z
−
1
x
2
x
Fig. B.1-1: Distance between 2 vectors
y
z

Appendix B page
2
2
2
1
(
)
n
j
j
j
y
z
y
z
=
−
=
−
∑
It is sometimes more convenient to express distance as a vector product.
Vector Product
The product (“inner product” or “sumproduct”) of two
n
-dimensional vectors
y
and
z
is.
1
n
i
i
i
y z
y z
=
⋅
=
∑
From this definition, it follows that the distributive law for multiplying vectors is the
same as for numbers.
Distributive law for vector multiplication
(
)
a
b
c
a b
a c
⋅
+
=
⋅
+
⋅
Orthogonal Vectors
Two vectors,
x
and
p
are depicted below in 3-dimensional space.
Fig. B.1-2: Orthogonal Vectors
1
x
O
x
2
x
3
x
x
p
p
p
x
−

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*