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LA (1)

LA (1) - 1 General Vector Spaces 1.1 Vector space axioms...

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Unformatted text preview: 1. General Vector Spaces 1.1. Vector space axioms. Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule for assigning to each pair of vectors u,v ∈ V a unique vector u + v . By scalar multiplication we mean a rule for associating to each scalar k and each u ∈ V a unique vector ku . The set V together with these operations is called a vector space, provided the following properties hold for all u,v,w ∈ V and scalars k,l in some field K : (1) If u,v ∈ V , then u + v ∈ V . We say that V is closed under addition. (2) u + v = v + u . (3) ( u + v ) + w = u + ( v + w ). (4) V contains an object 0, called the zero vector, which satisfies u + 0 = u for every vector u ∈ V . (5) For each u ∈ V there exists an object- u such that u + (- u ) = 0. (6) If u ∈ V , and k ∈ K , then ku ∈ V . We say V is closed under scalar multiplication. (7) k ( u + v ) = ku + kv . (8) ( k + l ) u = ku + lu . (9) k ( lu ) = ( kl ) u . (10) 1u = u, where 1 is the identity in K . Remark 1.2 . The most important vector spaces are real vector spaces (for which K = R in the preceding definition), and complex vector spaces (where K is the complex numbers C ). 1.2. Subspaces, linear independence, span, basis. Definition 1.3. A nonempty subset W of a vector space V is called a subspace if W is closed under scalar multiplication and addition. Definition 1.4. A set M = { v 1 ,...,v s } of vectors in V is called linearly in- dependent, provided the only set { c 1 ,...,c s } of scalars which solve the equation c 1 v 1 + c 2 v 2 + ... + c s v s = 0 is c 1 = c 2 = ... = c s = 0. If M is not linearly independent then it is called linearly dependent. Definition 1.5. The span of a set of vectors M = { v 1 ,...,v s } is the set of all possible linear combinations of the members of M . Definition 1.6. A set of vectors in a subspace W of V is said to be a basis for V if it is linearly independent and its span is V . Definition 1.7. A vector space V is finite-dimensional if it has a basis with finitely many vectors, and infinite-dimensional otherwise. If V is finite-dimensional, then the dimension of V is the number of vectors in any basis; otherwise the dimension of V is infinite. 1 2 1.3. Examples. Illustration 1: Euclidean and Complex Spaces The most important examples of finite-dimensional vector spaces are n-dimensional Euclidean space R n , and n-dimensional complex space C n . Illustration 2 . An important example of an infinite-dimensional vector space is the space of real-valued functions which have n-th order continuous derivatives on all of R , which we denote by C n ( R ). Definition 1.8. Let f 1 ( x ) ,f 2 ( x ) ,...,f n ( x ) be elements of C ( n- 1) ( R ). The Wron- skian of these functions is the determinant whose n-th row contains the ( n- 1) derivatives of the functions, fl fl fl fl fl fl fl fl f 1 ( x ) f 2 ( x ) · · · f n ( x ) f 1 ( x ) f 2 ( x ) · · · f n...
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LA (1) - 1 General Vector Spaces 1.1 Vector space axioms...

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