Vector Spaces. distributive law u,v. Associative Law. 1 v v. Let 1 be the unit element in F, then

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2 Def: V be a set of elements with a binary operation + is defined. F be a field. A multiplication operator between a F and v V is also defined. The V is called a vector space over the field F if: V is a commutative group under + a F & v V a v V distributive law u,v V and a,b F Associative Law a ( u v) ( a b) v Let 1 be the unit element in F, then a u a v a v b v ( a b) v a ( b v) 1 v v 2

3 The elements of V are called vectors and the elements of the field F are called scalars. The addition on V is called a vector addition and the multiplication that combines a scalar in F and a vector in V is referred to as scalar multiplication (or product) The additive identity of V is denoted by 0. Property I. Let 0 be the zero element of the field F. For any vector v in V, 0 v = 0. Property II. For any scalar c in F, c 0 = 0. (Left as an exercise) 3

4 Property III. For any scalar c in F and any vector v in V, (-c) v = c (- v ) = -(c v) i.e., (-c) v or c (- v ) is the additive inverse of the vector c v. (Left as an exercise) ( 1 Consider an ordered sequence of n components, a0, a1,..., a n ), where each component ai is an element from the binary field GF(2) (i.e., ai = 0 or 1). This sequence is called an n-tuple over GF(2). n Since there are two choices for each ai, we can construct 2 distinct n-tuples. Let Vn denote this set. Now we define an addition + on V as following : For any u = ( u0, u1,..., u n 1) and v = ( v0, v1,..., v n 1) in V, u + v = u v, u v,..., u n v ) n ( n 1 n 4

5 where u i +v i is carried out in modulo-2 addition. Clearly, u + v is also an n-tuple over GF(2). Hence is closed under the addition. We can readily verify that is a commutative group under the addition defined by (2.27). (1) we see that all zero n-tuple 0 = (0, 0,,0) is the additive identity. For any v in, v + v = v v, v v,..., v n v ) = (0, 0,,0) = 0 ( n 1 Hence, the additive inverse of each n-tuples in is itself. Since modulo-2 addition is commutative and associative, the addition is also commutative and associative. Therefore, is a commutative group under the addition. (2) we defined scalar multiplication of an n-tuple v in 5

6 by an element a from GF(2) as follows : a ( v 0, v1,..., v n 1) = ( a v0, a v1,..., a vn 1) (2.28) where a v i is carried out in modulo-2 multiplication. Clearly, a ( v0, v1,..., v n 1) is also an n-tuple in. If a = 1, 1 ( v 0, v1,..., v n 1) = ( 1 v0,1 v1,...,1 vn 1) = ( v0, v1,..., v n 1) By (2.27) and (2.28), the set of all n-tuples over GF(2) forms a vector space over GF(2) 6

7 A set of vectors v 1, v 2,,v k in a vector space V over a field F is said to be linearly dependent if and only if there exit k scalars a 1, a 2,, a k from F, not all zeros, such that a 1 v 1 + a 2 v a k v k = 0 A set of vectors v 1, v 2,,v k is said to be linearly independent if it is not linearly dependent. That is, if v 1, v 2,,v k are linearly independent, then a 1 v 1 + a 2 v a k v k 0 unless a 1 = a 2 = = a k = 0. EX. The vectors ( ), ( ), and ( ) are linearly dependent since 1 ( ) + 1 ( ) + 1 ( ) = ( ) 7

8 Example Let n=2. The vector space V 2 of all 2-tuples over GF(2) consists of the following 4 vectors : (0 0) (0 1) (1 0) (1 1) The vector sum of (0 0) and (0 1) is (0 0) + (0 1) = ( ) = (0 1) Using the rule of scalar multiplication defined by (2.28), we get 0 (1 0) = ( ) = (0 0) 1 (1 1) = ( ) = (1 1) V being a vector space of all n-tuples over any field F, it may happen that a subset S of V is also a vector space over F. Such a subset is called a subspace of V. 8

9 Theorem 2.18 Let S be a nonempty subset of a vector space V over a field F. Then S is a subspace of V if the following conditions are satisfied : (1) For any two vectors u and v in S, u + v is also a vector in S. (2) For an element a in F and any vector u in S, a u is also in S. (pf). Conditions (1) and (2) say simply that S is closed under vector addition and scalar multiplication of V. Condition (2) ensures that, for any vector v in S, its additive inverse (-1) v is also in S. Then, v + (-1) v = 0 is also in S. Therefore, S is a subgroup of V. Since the vectors of S are also vectors of V, the associative and distributive laws must hold for S. Hence, S is a vector space over F and is a subspace of V. 9

10 Let v 1, v 2,,v k be k vectors in a vector space V over a field F. Let a 1, a 2,, a k be k scalars from F. The sum a 1 v 1 + a 2 v a k v k is called a linear combination of v 1, v 2,,v k. Clearly, the sum of two linear combinations of v 1, v 2,,v k, (a 1 v 1 + a 2 v a k v k ) + (b 1 v 1 + b 2 v b k v k ) = (a 1 +b 1 )v 1 + (a 2 +b 2 )v (a k +b k )v k is also a linear combination of v 1, v 2,,v k, and the product of a scalar c in F and a linear combination of v 1, v 2,,v k, c (a 1 v 1 + a 2 v a k v k ) = ( c a1 ) v1 ( c a2) v2... ( c a k ) vk is also a linear combination of v 1, v 2,,v k Theorem 2.19 Let v 1, v 2,,v k be k vectors in a vector space V over a field F. The set of all linear combinations of v 1, v 2,,v k forms a subspace of V. 10

11 However, ( ), ( ), and ( ) are linearly independent. A set of vectors is said to span a vector space V if every vector in V is a linear combination of the vectors in the set. In any vector space or subspace there exits at least one set B of linearly independent vectors which span the space. This set is called a basis (or base) of the vector space. The number of vectors in a basis of a vector space is called the dimension of the vector space. (Note that the number of vectors in any two bases are the same.) 11

12 Consider the vector space of all n-tuples over GF(2). Let us form the following n n-tuples : ( ) e 0 e 1 e n- 1 ( ) ( ), where the n-tuple e i has only nonzero component at ith position. Then every n-tuple ( a0, a1,..., a n 1) in can be expressed as a linear combination of e 0, e 1,,e n-1 as follows : ( a0, a1,..., an 1) a0e0 a1e1... an 1en 1 12

13 Therefore, e0, e1,,en-1 span the vector space of all n- tuples over GF(2). We also see that e0, e1,,en-1 are linearly independent. Let u = ( u 0, u1,..., u n 1) and v = ( v0, v1,..., v n 1) be two n- tuples in. We define the inner product (or dot product) of u and v as u v u0v0 u1v1... u n 1vn 1, where ui viand ui vi + ui+1 vi+1 are carried out in modulo-2 multiplication and addition. Hence the inner product u v is a scalar in GF(2). If u v = 0, u and v are said to be orthogonal to each other. The inner product has the following properties : u v = v u u (v+w) = u v + u w (au) v = a(u v) 13

14 Let S be a k-dimension subspace of and let S d be the set of vectors in such that, for any u in S and v in S d, u v = 0. The set S d contains at least the all-zero n-tuple 0 = (0, 0,, 0), since for any u in S, 0 u = 0. Thus, S d is nonempty. For any element a in GF(2) and any v in S d, Therefore, a v is also in S d. Let v and w be any two vectors in S d. For any vector u in S, u (v+w) = u v + u w = = 0. This says that if v and w are orthogonal to u, the vector sum v + w is also orthogonal to u. Consequently, v + w is a vector in S d. It follows from Theorem 2.18 that S d is also a subspace of. This subspace is called the null (or dual) space of S. Conversely, S is also the null space of S d. 0 if a 0 a v v if a 1 14

15 Theorem 2.20 Let S be a k-dimension subspace of the vector space of all n-tuples over GF(2). The dimension of its null space S d is n-k. In other words, dim(s) + dim(s d )= n. 15