Definition 1. A set V is a vector space over the scalar field F {R, C} iff. there are two operations defined on V, called vector addition

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1 6 Vector Spaces with Inned Product Basis and Dimension Section Objective(s): Vector Spaces and Subspaces Linear (In)dependence Basis and Dimension Inner Product 6 Vector Spaces and Subspaces Definition A set V is a vector space over the scalar field F {R, C} iff there are two operations defined on V, called vector addition and scalar multiplication with the following properties: For all u, v, w V the vector addition satisfies (A) u + v V (closure of V ); (A) u + v = v + u (commutativity); (A3) (u + v) + w = u + (v + w) (associativity); (A4) V : + u = u, u V (existence of a neutral element); (A5) u V, ( u) V : u + ( u) = (existence of an opposite element) Furthermore, for all a, b F the scalar multiplication satisfies (M) au V (closure of V ); (M) u = u (multiplicative identity of F); (M3) a(bu) = (ab)u (associativity); (M4) a(u + v) = au + av (distributivity); (M5) (a + b)u = au + bu (distributivity) Example : One can show that V = R n is a vector space with vector addition and scalar multiplication defined by u v u v and u + v = u n au = a + u u u n v n = = au au au n u + v u + v u n + v n Find the neutral element (additive identity, guaranteed in (A4) ) and given vector v with components v i, i =,, n, determine its opposite element postulated by (A5)

2 Example : The space of m n matrices with real entries is a vector space given the standard addition and scalar multiplication of matrices Find the neutral element (additive identity, guaranteed in (A4)) and verify (M) Example 3: Is the set of polynomials of degree exactly 3 (with standard addition and scalar multiplication) a vector space? How about the set of polynomials of degree less than or equal to 3?

3 3 Definition subspace of V The subset W V of a vector space V over the field of scalars F is called a iff for all u, v W and all a, b F, the following holds true au + bv W Example 4: Which of the following sets W are subspaces of the vector space V? { ( ) } () V = R u, W = u = u u = () V = R, W = { ( u u = u ) } u = (3) V = R, W = { ( ) } u u = u u + u = (4) V = R 3, W = u = u u u = u 3 u 3 (5) V is the vector space of polynomials of degree at most 5 and W is the set of polynomials of degree at most

4 4 6 Linear Dependence and Independence Definition 3 The span of a finite set S = {u,, u n } in a vector space V over the field of scalars F, denoted as Span(S), is the set given by Span(S) = {u V u = c u + + c n u n, where c,, c n F} Theorem Given a finite set S in a vector space V, Span(S) is a subspace of V Proof: Since Span(S) contains all possible linear combinations of the elements in S, then Span(S) is closed under linear combinations This establishes the Theorem Example 5: Give a geometric description of the following () Span({v, v }) in R 3, where v = and v = () Span({v, v }) in R 3, where v = and v =

5 5 Definition 4 A finite set of vectors {v,, v k } in a vector space is called linearly dependent iff there exists a set of scalars {c,, c k }, not all of them zero, such that, c v + + c k v k = On the other hand, the {v,, v k } is called linearly independent iff Eq (*) implies that every scalar vanishes, ie, c = = c k = Example 6: Determine if the following sets are linearly independent and justify your claim (),, 3 (),, 3

6 6 63 Basis and Dimension Definition 5 A finite set S V is called a basis of the vector space V iff () S is linearly independent and () Span(S) = V Example 7: Determine if the following sets provide bases for the given vector space () V = R, S =,, 3 () V = R, S =, (3) V = R 3, S =,, 3

7 7 Definition 6 A vector space V is finite dimensional iff V has a finite basis or V is one of the following two extreme cases: V = or or V = {} Otherwise, the vector space V is called infinite dimensional Example 8: Give and example of an infinite dimensional vector space Theorem The number of elements in any basis of a finite dimensional vector space, V, is the same as in any other basis of V Example 9: Give an example of two different bases of R Definition 7 The dimension of a finite dimensional vector space V with a finite basis, denoted as dim(v ), is the number of elements in any basis of V The extreme cases of V = and V = {} are defined as zero dimensional Example : Determine the dimension of the vector space consisting of all polynomials of degree at most

8 8 Definition 8a The dot product on R n is the function R n R n R, defined by u u v v u n v n = u v + u v + u n v n The dot product norm of a vector v R n is the function : R n R, defined by v = v v The angle between vectors u, v R n is the number θ [, π], such that cos θ = u v u v Thus, we say two vectors u, v R n are orthogonal if u v = How do we define a dot product on C n? Note, we want the norm of a vector the be associated with its distance from the zero vector, so the norm should be a non-negative real number Definition 8b The dot product on R n is the function : R n R n R, defined by u u v v u n v n = u v + u v + u n v n Example : Find the norm of the vector + i C

9 9 Theorem 3 The dot product on F n, with n, satisfies for all vectors x, y, z F n, and all scalars a, b F, the following properties: (a) x y = y x, (Symmetry F = R); (a) x y = y x, (Conjugate symmetry, for F = C); (b) (ax + by) z = a(x z) + b(y z), (Linearity on the st argument); (c) x x, and x x = iff x =, (Positive definiteness) Since we saw how the dot product induces a norm and therefore a distance between two vectors in a vector space, can we find a similar notion for a more general vector space than F n? Definition 9 Let V be a vector space over the scalar field F {R, C} A function, : V V F is called an inner product iff for every x, y, z V and every a, b F the function, satisfies: (a) x y = y x, (a) x y = y x, (b) (ax + by) z = a(x z) + b(y z), (c) x x, and x x = iff x =, (Symmetry F = R); (Conjugate symmetry, for F = C); (Linearity on the st argument); (Positive definiteness) An inner product space is a pair (V,, ) of a vector space with an inner product Example : Let P n denote the space of polynomials with real coefficients of degree at most n Does the following function from P n P n R satisfy the conditions of an inner product? For p, q P n, p, q = p(x)q(x) dx