Linear Algebra

76 Chapter Two. Vector SpacesIDefinition of Vector SpaceWe shall study structures with two operations, an addition and a scalar multiplication,that are subject to some simple conditions. We will reflect more onthe conditions later, but on first reading notice how reasonable they are. Forinstance, surely any operation that can be called an addition (e.g., column vectoraddition, row vector addition, or real number addition) will satisfy conditions(1) through (5) below.I.1 Definition and Examples1.1 Definition A vector space (over R) consists of a set V along with twooperations ‘+’ and ‘·’ subject to these conditions.Where ⃗v, ⃗w ∈ V, (1) their vector sum ⃗v+ ⃗w is an element of V. If ⃗u,⃗v, ⃗w ∈ Vthen (2) ⃗v + ⃗w = ⃗w + ⃗v and (3) (⃗v + ⃗w) + ⃗u = ⃗v + (⃗w + ⃗u). (4) There is a zerovector ⃗0 ∈ V such that ⃗v + ⃗0 = ⃗v for all ⃗v ∈ V. (5) Each ⃗v ∈ V has an additiveinverse ⃗w ∈ V such that ⃗w + ⃗v = ⃗0.If r, s are scalars, members of R, and ⃗v, ⃗w ∈ V then (6) each scalar multipler · ⃗v is in V. If r, s ∈ R and ⃗v, ⃗w ∈ V then (7) (r + s) · ⃗v = r · ⃗v + s · ⃗v, and(8) r · (⃗v + ⃗w) = r · ⃗v + r · ⃗w, and (9) (rs) · ⃗v = r · (s · ⃗v), and (10) 1 · ⃗v = ⃗v.1.2 Remark The definition involves two kinds of addition and two kinds ofmultiplication and so may at first seem confused. For instance, in condition (7)the ‘+’ on the left is addition between two real numbers while the ‘+’ on the rightrepresents vector addition in V. These expressions aren’t ambiguous because,for example, r and s are real numbers so ‘r + s’ can only mean real numberaddition.The best way to go through the examples below is to check all ten conditionsin the definition. We write that check out at length in the first example. Use itas a model for the others. Especially important are the closure conditions, (1)and (6). They specify that the addition and scalar multiplication operations arealways sensible — they are defined for every pair of vectors and every scalar andvector, and the result of the operation is a member of the set (see Example 1.4).1.3 Example The set R 2 is a vector space if the operations ‘+’ and ‘·’ have theirusual meaning.( ) ( ) ( ) ( ) ( )x 1 y 1 x 1 + y 1 x 1 rx 1+ =r · =x 2 y 2 x 2 + y 2 x 2 rx 2We shall check all of the conditions.There are five conditions in the paragraph having to do with addition. For(1), closure of addition, note that for any v 1 , v 2 , w 1 , w 2 ∈ R the result of the