Abstract Algebra Theory and Applications - Computer Science ...

18.3 LINEAR INDEPENDENCE 315Let V be any vector field over a field F and suppose that v 1 , v 2 , . . . , v nare vectors in V and α 1 , α 2 , . . . , α n are scalars in F . Any vector w in V ofthe formn∑w = α i v i = α 1 v 1 + α 2 v 2 + · · · + α n v ni=1is called a linear combination of the vectors v 1 , v 2 , . . . , v n . The spanningset of vectors v 1 , v 2 , . . . , v n is the set of vectors obtained from all possible linearcombinations of v 1 , v 2 , . . . , v n . If W is the spanning set of v 1 , v 2 , . . . , v n ,then we often say that W is spanned by v 1 , v 2 , . . . , v n .Proposition 18.2 Let S = {v 1 , v 2 , . . . , v n } be vectors in a vector space V .Then the span of S is a subspace of V .Proof. Let u and v be in S. We can write both of these vectors as linearcombinations of the v i ’s:Thenu = α 1 v 1 + α 2 v 2 + · · · + α n v nv = β 1 v 1 + β 2 v 2 + · · · + β n v n .u + v = (α 1 + β 1 )v 1 + (α 2 + β 2 )v 2 + · · · + (α n + β n )v nis a linear combination of the v i ’s. For α ∈ F ,αu = (αα 1 )v 1 + (αα 2 )v 2 + · · · + (αα n )v nis in the span of S.□18.3 Linear IndependenceLet S = {v 1 , v 2 , . . . , v n } be a set of vectors in a vector space V . If thereexist scalars α 1 , α 2 . . . α n ∈ F such that not all of the α i ’s are zero andα 1 v 1 + α 2 v 2 + · · · + α n v n = 0,then S is said to be linearly dependent. If the set S is not linearly dependent,then it is said to be linearly independent. More specifically, S is alinearly independent set ifimplies thatfor any set of scalars {α 1 , α 2 . . . α n }.α 1 v 1 + α 2 v 2 + · · · + α n v n = 0α 1 = α 2 = · · · = α n = 0