Linear Algebra

Section I. Definition of Vector Space 85
1.9 Example The set {f � � f : N → R} of all real-valued functions of one natural
number variable is a vector space under the operations
(f1 + f2)(n) =f1(n)+f2(n) (r · f)(n) =rf(n)
so that if, for example, f1(n) =n 2 +2sin(n) andf2(n) =− sin(n) +0.5 then
(f1 +2f2)(n) =n 2 +1.
We can view this space as a generalization of Example 1.3 by thinking of
these functions as “the same” as infinitely-tall vectors:
n f(n) =n 2 +1
0 1
1 2
2 5
3 10
.
.
corresponds to
⎛ ⎞
1
⎜ 2 ⎟
⎜ 5 ⎟
⎜
⎝
10⎟
⎠
.
with addition and scalar multiplication are component-wise, as before. (The
“infinitely-tall” vector can be formalized as an infinite sequence, or just as a
function from N to R, in which case the above correspondence is an equality.)
1.10 Example The set of polynomials with real coefficients
{a0 + a1x + ···+ anx n � � n ∈ N and a0,... ,an ∈ R}
makes a vector space when given the natural ‘+’
(a0 + a1x + ···+ anx n )+(b0 + b1x + ···+ bnx n )
and ‘·’.
=(a0 + b0)+(a1 + b1)x + ···+(an + bn)x n
r · (a0 + a1x + ...anx n )=(ra0)+(ra1)x + ...(ran)x n
This space differs from the space P3 of Example 1.8. This space contains not just
degree three polynomials, but degree thirty polynomials and degree three hundred
polynomials, too. Each individual polynomial of course is of a finite degree,
but the set has no single bound on the degree of all of its members.
This example, like the prior one, can be thought of in terms of infinite-tuples.
For instance, we can think of 1 + 3x +5x 2 as corresponding to (1, 3, 5, 0, 0,...).
However, don’t confuse this space with the one from Example 1.9. Each member
of this set has a bounded degree, so under our correspondence there are no
elements from this space matching (1, 2, 5, 10, ...). The vectors in this space
correspond to infinite-tuples that end in zeroes.
1.11 Example The set {f � � f : R → R} of all real-valued functions of one real
variable is a vector space under these.
(f1 + f2)(x) =f1(x)+f2(x) (r · f)(x) =rf(x)
The difference between this and Example 1.9 is the domain of the functions.