Vector Space 2.13 2001/12/30 2005/10/18 16:34:11.747 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Elizabeth Chan lizychan@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu basis direct sum linear independence span subspace vector This module introduces vector space. A vector space consists of the following four elements: A set of vectors V, A field of scalars  (where, for our purposes,  is either or ), The operations of vector addition "+" (i.e., + : V V V ) The operation of scalar multiplication "⋅"(i.e., ⋅ :  V V ) for which the following properties hold. (Assume x y z V and α β  .) Properties Examples commutativity x y y x associativity x y z x y z α β x α β x distributivity α ⋅ x y ⋅ α x ⋅ α y α β x α x β x additive identity x V 0 0 V x 0 x additive inverse x V x x V x x 0 multiplicative identity x V ⋅ 1 x x Important examples of vector spaces include Properties Examples real N-vectors V N ,  complex N-vectors V N ,  sequences in " lp " V x n n n x n p ,  functions in " ℒp " V f t t f t p ,  where we have assumed the usual definitions of addition and multiplication. From now on, we will denote the arbitrary vector space (V, , +, ⋅) by the shorthand V and assume the usual selection of (, +, ⋅). We will also suppress the "⋅" in scalar multiplication, so that ⋅ α x becomes α x . A subspace of V is a subset M V for which x y x M y M x y M x M α  α x M Note that every subspace must contain 0, and that V is a subspace of itself. The span of set S V is the subspace of V containing all linear combinations of vectors in S. When S x0 … x N - 1 , ≔ span S i 0 N 1 α i x i α i  A subset of linearly-independent vectors x0 … x N - 1 V is called a basis for V when its span equals V. In such a case, we say that V has dimension N. We say that V is infinite-dimensional The definition of an infinite-dimensional basis would be complicated by issues relating to the convergence of infinite series. Hence we postpone discussion of infinite-dimensional bases until the Hilbert Space section. if it contains an infinite number of linearly independent vectors. V is a direct sum of two subspaces M and N, written V M N , iff every x V has a unique representation x m n for m M and n N . Note that this requires M N 0