Page 18 - Linear Algebra Done Right

Chapter 1. Vector Spaces
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Definition of Vector Space
Before defining what a vector space is, let’s look at two important
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examples. The vector space R , which you can think of as a plane,
consists of all ordered pairs of real numbers:
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R ={(x, y) : x, y ∈ R}.
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The vector space R , which you can think of as ordinary space, consists
of all ordered triples of real numbers:
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R ={(x,y,z) : x, y, z ∈ R}.
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To generalize R and R to higher dimensions, we first need to dis-
cuss the concept of lists. Suppose n is a nonnegative integer. A list of
length n is an ordered collection of n objects (which might be num-
bers, other lists, or more abstract entities) separated by commas and
Many mathematicians surrounded by parentheses. A list of length n looks like this:
call a list of length n an
n-tuple. (x 1 ,...,x n ).

Thus a list of length 2 is an ordered pair and a list of length 3 is an
ordered triple. For j ∈{1,...,n}, we say that x j is the j th coordinate
of the list above. Thus x 1 is called the first coordinate, x 2 is called the
second coordinate, and so on.
Sometimes we will use the word list without specifying its length.
Remember, however, that by definition each list has a finite length that
is a nonnegative integer, so that an object that looks like

(x 1 ,x 2 ,...),

which might be said to have infinite length, is not a list. A list of length
0 looks like this: (). We consider such an object to be a list so that
some of our theorems will not have trivial exceptions.
Two lists are equal if and only if they have the same length and
the same coordinates in the same order. In other words, (x 1 ,...,x m )
equals (y 1 ,...,y n ) if and only if m = n and x 1 = y 1 ,...,x m = y m .
Lists differ from sets in two ways: in lists, order matters and repeti-
tions are allowed, whereas in sets, order and repetitions are irrelevant.
For example, the lists (3, 5) and (5, 3) are not equal, but the sets {3, 5}
and {5, 3} are equal. The lists (4, 4) and (4, 4, 4) are not equal (they