Abstract Algebra Theory and Applications - Computer Science ...

18.1 DEFINITIONS AND EXAMPLES 313multiplied. In general, it is only possible to multiply a vector with a scalar.To differentiate between the scalar zero and the vector zero, we will writethem as 0 and 0, respectively.Let us examine several examples of vector spaces. Some of them will bequite familiar; others will seem less so.Example 1. The n-tuples of real numbers, denoted by R n , form a vectorspace over R. Given vectors u = (u 1 , . . . , u n ) and v = (v 1 , . . . , v n ) in R n andα in R, we can define vector addition byu + v = (u 1 , . . . , u n ) + (v 1 , . . . , v n ) = (u 1 + v 1 , . . . , u n + v n )and scalar multiplication byαu = α(u 1 , . . . , u n ) = (αu 1 , . . . , αu n ).Example 2. If F is a field, then F [x] is a vector space over F . The vectorsin F [x] are simply polynomials. Vector addition is just polynomial addition.If α ∈ F and p(x) ∈ F [x], then scalar multiplication is defined by αp(x). Example 3. The set of all continuous real-valued functions on a closedinterval [a, b] is a vector space over R. If f(x) and g(x) are continuous on[a, b], then (f + g)(x) is defined to be f(x) + g(x). Scalar multiplication isdefined by (αf)(x) = αf(x) for α ∈ R. For example, if f(x) = sin x andg(x) = x 2 , then (2f + 5g)(x) = 2 sin x + 5x 2 .Example 4. Let V = Q( √ 2 ) = {a + b √ 2 : a, b ∈ Q}. Then V is a vectorspace over Q. If u = a+b √ 2 and v = c+d √ 2, then u+v = (a+c)+(b+d) √ 2is again in V . Also, for α ∈ Q, αv is in V . We will leave it as an exercise toverify that all of the vector space axioms hold for V .Proposition 18.1 Let V be a vector space over F . Then each of the followingstatements is true.1. 0v = 0 for all v ∈ V .2. α0 = 0 for all α ∈ F .3. If αv = 0, then either α = 0 or v = 0.4. (−1)v = −v for all v ∈ V .