Examples of vector spaces

The set P of all polynomials in the variable x. This is the set of all expressions of the form

P(x) = anxn + an–1xn–1 + . . . + a2x2 + a1x + a0

where the coefficients an, an–1, ..., a2, a1, a0 are all constants. The zero polynomial is the one with all coefficients zero, and the negative of any polynomial is the one whose coefficients are the negatives of the coefficients of the original – for example, the negative of the polynomial

P(x) = 6x4 + 2x3 – 5x2 – 7x + 11

is the polynomial

(–P)(x) = –6x4 – 2x3 + 5x2 + 7x – 11.

The rest of the ten rules are just basic calculation rules for polynomials you learned in algebra.

The set F of all real-valued functions, that is, the set of all functions f:R → R which take real numbers into real numbers. Addition and scalar multiplication on F are defined by

(f + g)(x) = f(x) + g(x)        and        (cf)(x) = cf(x)

for all real numbers x. The zero vector of F is the function that takes all x's into 0, and the negative of any real-valued function f is the function given by (–f)(x) = –f(x) for all x. So for example, if f(x) = sin x and g(x) = x2, then (f + g)(x) = sin x + x2 and (–f)(x) = –sin x.

Here's an example of a "weird" vector space. Remember that the addition and scalar multiplication operations don't have to look like "normal" operations as long as they obey the ten rules for vector space operations. Suppose our set W consists of all ordered pairs (p, q) of real numbers with p + q = 1. Define addition and scalar multiplication as follows:

(p, q) + (r, s) = (p + r – 1, q + s)    and    c•(p, q) = (cp – c + 1, cq).

(The + sign and the • are red to warn you that these are not the usual operations of addition and scalar multiplication.) The set W with these operations turns out to be a vector space, though its operations look nothing like any other "normal" operations. To show that W is indeed a vector space, you have to check that this addition and scalar multiplication, strange as they are, still obey all of the ten rules. A few hints to start: the zero vector is (1, 0), and the negative of any vector (p, q) is (2 – p, –q).