Linear Algebra

Section I. Definition of Vector Space 811.10 Example The set {f ∣ ∣ f: N → R} of all real-valued functions of one naturalnumber variable is a vector space under the operations(f 1 + f 2 ) (n) = f 1 (n) + f 2 (n) (r · f) (n) = r f(n)so that if, for example, f 1 (n) = n 2 + 2 sin(n) and f 2 (n) = − sin(n) + 0.5 then(f 1 + 2f 2 ) (n) = n 2 + 1.We can view this space as a generalization of Example 1.3 — instead of 2-tallvectors, these functions are like infinitely-tall vectors.n f(n) = n 2 + 10 11 22 53 10..corresponds to⎛ ⎞125⎜⎝10⎟⎠.Addition and scalar multiplication are component-wise, as in Example 1.3. (Wecan formalize “infinitely-tall” by saying that it means an infinite sequence, orthat 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 a0 , . . . , 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 )and ‘·’.= (a 0 + b 0 ) + (a 1 + b 1 )x + · · · + (a n + b n )x nr · (a 0 + a 1 x + . . . a n x n ) = (ra 0 ) + (ra 1 )x + . . . (ra n )x nThis space differs from the space P 3 of Example 1.8. This space containsnot just degree three polynomials, but degree thirty polynomials and degreethree hundred polynomials, too. Each individual polynomial of course is of afinite 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 ofthe set has a finite degree, that is, under the correspondence there is no elementfrom this space matching (1, 2, 5, 10, . . . ). Vectors in this space correspond toinfinite-tuples that end in zeroes.1.12 Example The set {f ∣ ∣ f: R → R} of all real-valued functions of one realvariable is a vector space under these.(f 1 + f 2 ) (x) = f 1 (x) + f 2 (x) (r · f) (x) = r f(x)The difference between this and Example 1.10 is the domain of the functions.