16. Systems of Linear Equations 1 Matrices and Systems of Linear ...

March 31, 2013 16-256.2 Subspaces of R n , Null Space and Range of a LinearMapIt is convenient to have name for subsets of R n which behave well undervector addition and scalar multiplication.Definition. A subset W of R n is called a linear subspace (or, simply asubspace) if it satisfies the following two properties.1. For any two vectors v and w in W , we have v + w ∈ W .2. For any vector v ∈ W and scalar α, we have αv ∈ W .We often say that W is closed under vector addition and scalar multiplication.Let W be a subspace of R n which contains at least one non-zero vector.A basis for W is a maximal linear independent set B = {v 1 , v 2 , . . . , v k } ofvectors in W . This means that, for any vector w in W the setB 1 = {w} ⋃ Bis no longer linear independent. That is, we cannot increase B inside Wand keep it a linearly independent set.It is a fact that any two bases of a subspace W have the same numberof elements. This common number is called the dimension of W . Note thesingle element subset {0} is a subspace of R n . We define its dimension to be0.Let B be any subset of R n . Then, the subspace spanned by B, denotedsp(B) is defined to the set of finite linear combinations a 1 v 1 + a 2 v 2 + a j v jwhere a i is a scalar and v i is a vector in B for all i.If B = {v 1 , v 2 , . . . , v k } is a finite set, then we write sp(v 1 , v 2 , . . . , v k ) forsp(B).Note that if dim(sp(B)) = d and k > d, then the set B cannot be linearlyindependent.Examples.1. The subspaces of R 2 consist of(a) the set {0|} consisting of the zero vector. (dimension 0)(b) the lines through the origin (dimension 1)