Page 19 - Linear Algebra Done Right

Definition of Vector Space
do not have the same length), though the sets {4, 4} and {4, 4, 4} both
equal the set {4}.
3
2
To define the higher-dimensional analogues of R and R , we will 5
simply replace R with F (which equals R or C) and replace the 2 or 3
with an arbitrary positive integer. Specifically, fix a positive integer n
for the rest of this section. We define F n to be the set of all lists of
length n consisting of elements of F:
n
F ={(x 1 ,...,x n ) : x j ∈ F for j = 1,...,n}.
For example, if F = R and n equals 2 or 3, then this definition of F n
2
3
agrees with our previous notions of R and R . As another example,
4
C is the set of all lists of four complex numbers:
4
C ={(z 1 ,z 2 ,z 3 ,z 4 ) : z 1 ,z 2 ,z 3 ,z 4 ∈ C}.
n
If n ≥ 4, we cannot easily visualize R as a physical object. The same For an amusing
1
problem arises if we work with complex numbers: C can be thought account of how R 3
of as a plane, but for n ≥ 2, the human brain cannot provide geometric would be perceived by
2
n
models of C . However, even if n is large, we can perform algebraic a creature living in R ,
3
2
n
manipulations in F as easily as in R or R . For example, addition is read Flatland: A
n
defined on F by adding corresponding coordinates: Romance of Many
Dimensions, by Edwin
1.1 (x 1 ,...,x n ) + (y 1 ,...,y n ) = (x 1 + y 1 ,...,x n + y n ). A. Abbott. This novel,
published in 1884, can
Often the mathematics of F n becomes cleaner if we use a single help creatures living in
entity to denote an list of n numbers, without explicitly writing the three-dimensional
n
coordinates. Thus the commutative property of addition on F should space, such as
be expressed as ourselves, imagine a
x + y = y + x physical space of four
or more dimensions.
n
for all x, y ∈ F , rather than the more cumbersome
(x 1 ,...,x n ) + (y 1 ,...,y n ) = (y 1 ,...,y n ) + (x 1 ,...,x n )

for all x 1 ,...,x n ,y 1 ,...,y n ∈ F (even though the latter formulation
is needed to prove commutativity). If a single letter is used to denote
n
an element of F , then the same letter, with appropriate subscripts,
is often used when coordinates must be displayed. For example, if
n
x ∈ F , then letting x equal (x 1 ,...,x n ) is good notation. Even better,
work with just x and avoid explicit coordinates, if possible.