Copyright © Paul Kantor, 2011.
Today we are going to move from vectors to some concepts that are very important for
searching and other things that computers can do with texts. These things are called
stems or roots of words, and they are the first step towards having computers ‘recognize
concepts’ in texts.
(0)
A note about the “cheat sheet” that you can take to exams. It should always include
the tables of logarithms and trigonometric functions. And when you copy formulas
into it, be very careful to copy themcorrectly, or they will not be much use.
Now for the material.
1
Dimensions, Components and Bases.
.
Let’s recall that a vector is an indexed array of numbers. For discussion, we have to write
them in some orcer, and so, on the blackboard, or kin the notes, a vector is actually an
ordered set of numbers. For example, a=(1,3,5) is a vector, and so is b=(72, -3, -4, 6).
The individual numebr are called the components of the vectors.
. So, for example, “3” is
a component of the first vector, and –3 is a component of the second vector. In writing
about them we use some subscript (usually
i or j or k
) to represent the position that the
component occupies. So we would write that
2
3
a
or
2
3,
b
and so forth,
Clearly the
number
of components that the vector has is important.
This number is called
the
dimensionality
or the
dimension
of the vector.
It is sometimes written as
Dim(v)
where
v
is the name of the vector. So
Dim(a)=3; Dim(b)=4.
What is the dimensionality
Dim(c),
when
c=
(2,3,0,5,0,4)? The answer is 6 (not 4)
because the zeroes are numbers, and they count.
Addition and dimension. We recall that we can add vectors by adding the corresponding
componenets. For example, the sum of a=(1,2,3) and b=(3,2,1) is just (4,4,4). But lets go
back to our example of
, a=(1,3,5) is a vector, and so is b=(72, -3, -4, 6). Can we add
them? The answer is
NO
!. They have different numbers of components, and so adding
them is
not even defined.
Sometimes people think that we could pad a by putting another
0 in it. But that is simply not allowed (we would not know whether to put it first, or last,
or somewhere in the middle, if we tried.).
1.1 What do the components mean?
As pure mathematicians, we can happily manipulate vectors, without worrying about
what they meamn. That is one of the great powers of mathematics as a tool. But the fact
that they do meansomehting, in particular cases, tells us to be careful about how we use
them.