Linear Algebra, 2020a

90 Chapter Two. Vector Spaces
Addition and scalar multiplication are component-wise, as in Example 1.4. (We
can formalize “infinitely-tall” by saying that it means an infinite sequence, or
that it means a function from N to R.)
1.11 Example The set of polynomials with real coefficients
{a 0 + a 1 x + ···+ a n x n | n ∈ N and a 0 ,...,a n ∈ R}
makes a vector space when given the natural ‘+’
(a 0 + a 1 x + ···+ a n x n )+(b 0 + b 1 x + ···+ b n x n )
=(a 0 + b 0 )+(a 1 + b 1 )x + ···+(a n + b n )x n
and ‘·’.
r · (a 0 + a 1 x + ...a n x n )=(ra 0 )+(ra 1 )x + ...(ra n )x n
This space differs from the space P 3 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.
We can think of this example, like the prior one, in terms of infinite-tuples.
For instance, we can think of 1 + 3x + 5x 2 as corresponding to (1,3,5,0,0,...).
However, this space differs from the one in Example 1.10. Here, each member of
the set has a finite degree, that is, under the correspondence there is no element
from this space matching (1, 2, 5, 10, . . . ). Vectors in this space correspond to
infinite-tuples that end in zeroes.
1.12 Example The set {f | f: R → R} of all real-valued functions of one real
variable is a vector space under these.
(f 1 + f 2 )(x) =f 1 (x)+f 2 (x) (r · f)(x) =rf(x)
The difference between this and Example 1.10 is the domain of the functions.
1.13 Example The set F = {a cos θ + b sin θ | a, b ∈ R} of real-valued functions of
the real variable θ is a vector space under the operations
(a 1 cos θ + b 1 sin θ)+(a 2 cos θ + b 2 sin θ) =(a 1 + a 2 ) cos θ +(b 1 + b 2 ) sin θ
and
r · (a cos θ + b sin θ) =(ra) cos θ +(rb) sin θ
inherited from the space in the prior example. (We can think of F as “the same”
as R 2 in that a cos θ + b sin θ corresponds to the vector with components a and
b.)