Page 96 - 35Linear Algebra

96 Vectors in Space, n-Vectors

Example 55 Consider the set S = {∗, ?, #} from chapter 1 review problem 9. A
S
particular element of R is the function a explicitly defined by
?
#
∗
a = 3, a = 5, a = −2.
It is not natural to write
   
3 −2
a =  5  or a =  3 
−2 5
because the elements of S do not have an ordering, since as sets {∗, ?, #} = {?, #, ∗}.

3
S
In this important way, R seems different from R . What is more evident
are the similarities; since we can add two functions, we can add two elements
S
of R :
{∗,?,#}
Example 56 Addition in R
{∗,?,#}
If a, b ∈ R such that
∗
?
#
a = 3, a = 5, a = −2
and
#
?
∗
b = −2, b = 4, b = 13
S
then a + b ∈ R is the function such that
∗
#
?
(a + b) = 3 − 2 = 1, (a + b) = 5 + 4 = 9, (a + b) = −2 + 13 = 11 .
Also, since we can multiply functions by numbers, there is a notion of
S
scalar multiplication on R .
S
Example 57 Scalar Multiplication in R
{∗,?,#}
If a ∈ R such that
?
∗
#
a = 3, a = 5, a = −2
{∗,?,#}
then 3a ∈ R is the function such that
∗
?
#
(3a) = 3 · 3 = 9, (3a) = 3 · 5 = 15, (3a) = 3(−2) = −6 .
2
3
We visualize R and R in terms of axes. We have a more abstract picture
5
4
n
S
of R , R and R for larger n while R seems even more abstract. However,
S
when thought of as a simple “shopping list”, you can see that vectors in R
in fact, can describe everyday objects. In chapter 5 we introduce the general
definition of a vector space that unifies all these different notions of a vector.
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