Page 95 - 35Linear Algebra

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4.4 Vectors, Lists and Functions: R 95

There is no information of ordering here and no information about how many
carrots you will buy. This set by itself is not a vector; how would we add
such sets to one another?
If you were a more careful shopper your list might look like the following.

What you have really done here is assign a number to each element of the
set S. In other words, the second list is a function
f : S −→ R .

Given two lists like the second one above, we could easily add them – if you
plan to buy 5 apples and I am buying 3 apples, together we will buy 8 apples!
In fact, the second list is really a 5-vector in disguise.
In general it is helpful to think of an n-vector as a function whose domain
is the set {1, . . . , n}. This is equivalent to thinking of an n-vector as an
ordered list of n numbers. These two ideas give us two equivalent notions for
the set of all n-vectors:
 

a 1
 
 
n
.
R :=  .  1 n = {a : {1, . . . , n} → R} =: R {1,··· ,n}
 .  a , . . . , a ∈ R
 n 
a
 
{1,··· ,n}
The notation R is used to denote the set of all functions from {1, . . . , n}
to R.
S
Similarly, for any set S the notation R denotes the set of functions from
S to R:
S
R := {f : S → R} .
When S is an ordered set like {1, . . . , n}, it is natural to write the components
in order. When the elements of S do not have a natural ordering, doing so
might cause confusion.

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