math111_logo Vector Space

3. Definition

The definition of vector spaces is based on the addition and scalar multiplications, which were used in the expressions u + c1v1 + c2v2, u(t) + c1v1(t) + c2v2(t), U + c1V1 + c2V2.

A vector space is a set V, its elements u called vectors, with two operations

u + v: V×VV
scalar multiplication
cu: R×VV

satisfying the following properties

u + v = v + u
(u + v) + w = u + (v + w)
There is a special vector 0V such that u + 0 = u = 0 + u
For any uV there is a vector - u such that u + (- u) = 0
c(u + v) = cu + cv
(c + d)u = cu + du
(cd)u = c(du)
1u = u

The eight properties comes from the properties satisfied by the euclidean vectors. Moreover, the notation R×VV means that the scalar multiplication takes a (number, vector)-pair and produces a vector.

Briefly speaking, a vector space is any set up in which addition and scalar multiplication can be done in a reasonable way (as measured by the eight properties).

Example The euclidean space Rn, with the following operations

(x1, x2, ..., xn) + (y1, y2, ..., yn) = (x1 + y1, x2 + y2, ..., xn + yn),
c(x1, x2, ..., xn) = (cx1, cx2, ..., cxn),

satisfies the eight properties and is the primary example of vector spaces.

Example Let F(X) be all the real valued functions on a set X. For example, F(R) is all the real functions defined on the real line, and F[0,1] is all the real functions defined on the interval [0,1]. With the usual operations

(f + g)(t) = f(t) + g(t),
(cf)(t) = c(f(t)),

F(X) is a vector space. In fact, two functions can also be multiplied together. However, this operation is not needed for the vector space.

By imposing additional conditions on functions, we get

C(R) = all continuous functions,
C1(R) = all functions f(t) with continuous derivative f'(t),
Ck(R) = all functions with continuous k-th order derivative.

We also use C0(R) to denote C(R) and use C(R) to denote functions with derivatives of any order. With the same addition and scalar multiplication, these are also vector spaces. The important thing to verify here is that if f(t) and g(t) satisfy the additional conditions (say, continuous), then f(t) + g(t), cf(t) also satisfy the additional conditions (also continuous). The verification follows from the usual theorems in calculus.

Example Another special collection of functions is polynomials. For any n, let Pn be the collection of polynomials

p(t) = a0 + a1t + a2t2 + ... + antn,

of degree ≤ n. The usual addition and scalar multiplication

(a0 + a1t + a2t2 + ... + antn) + (b0 + b1t + b2t2 + ... + bntn) = (a0 + b0) + (a1 + b1)t + (a2 + b2)t2 + ... + (an + bn)tn,
c(a0 + a1t + a2t2 + ... + antn) = ca0 + ca1t + ca2t2 + ... + cantn,

are consistent with the similar operations for general functions, and have the eight properties. Therefore Pn is a vector space.

Example Given m and n, all the m by n matrices, together with the usual addition and scalar multiplication, form a vector space M(m, n). The special collection Sym(n) of symmetric matrices, with the same operations, is also a vector space. Similar to the case of special collections of functions, the key reason for Sym(n) to be a vector space is A and B symmetric ⇒ A + B and cA are symmetric.

Note that the additions and scalar multiplications in the examples have to be the routine one because otherwise some of the eight properties will fail.

Finally, in a vector space we may combine addition and scalar multiplication and form the linear combination c1u1 + c2u2 + ... + ckuk.

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