Linear spaces review - Linear Spaces we have seen(12.1-12.3...

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Linear Spaces we have seen (12.1-12.3 of Apostol) that n-tuple space has the following properties : V, Addition: 1. (Commutativity) A + B = B + A. 2. (Associativity) A + (B+c) = (A+B) + C. 3. (Existence of zero) There is an element - 0 such 'that A + - 0 = A for all A. 4. (Existence of negatives) Given A, there is a B such that A + B = - 0. Scalar multiplication: 5. (Associativity) c (dA) = (cd)A. 6. (Distributivity) (c+d)A = cA + dA, c(A+B) = cA + cB. 7. (Multiplication by unity) 1A = A. Definition. More generally, let V be any set of objects (which we call vectors). And suppose there are two operations on V, as follows: The first is an operation (denoted +) that assigns to each pair A, B of vectors, a vector denoted A + B. The second is an operation that assigns to each real number c and each vector A, a vector denoted cA. Suppose also that the seven preceding properties hold. Then V, with these two opera- tions, is called a linear space (or a vector space). The seven properties are called the axioms -- for a linear space.
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There are many examples of linear spaces besides n--tuple space i ' n The study of linear spaces and their properties is dealt with in a subject called Linear Algebra. WE! shall treat only those aspects of linear algebra needed for calculus. Therefore we will be concerned only with n-tuple space and with certain of its subsets called "linear subspaces" : Vn - Definition. Let W be a non-empty subset of Vn ; suppose W is closed under vector addition and scalar multiplication. Then W is called a linear subspace of Vn (or sometimes simply a subspace of Vn .) To say W is closed under vector addition and scalar multiplication means that for every pair A, B of vectors of W, and every scalar c, the vectors A + B a~d cA belong to W. Note that it is automatic that the zero vector Q belongs to W, since for any A I W, we have Q = OA. Furthermore, for each A in W, the vector -A is also in W. This means (as you can readily check) that W is a linear space in its own right (i.e., f . it satisfies all the axioms for a linear,space). S~bspaces of my be specified in many different ways, as we shall Vn see. Example 1. The subset of Vn consisting of the 9-tuple alone is a subspace of it is ths "smallest possible" sub- Vn; space. Pad of course V, is by d e f i n i t i o n a subspace of Vn; it is the " l a r g e s t possible" subspace. W;ample 2. Let A be a fixed non-zero vector, The subset of Vn consisting of all vectors X of the form X = cA is a subspace of . ' n It is called the subspace spanned by A. In the case n = 2 or 3, it can be pictured as consisting of all vectors lying on a line through the origin.
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Example 3. Let A and B be given non-zero vectors that are not 1 parallel. The subset of Vn consisting of all vectors of the form is a subspace of V It is called the subspace spanned by A and no B. In the case n = 3, it can be pictured as consisting of all vectors lying in the plane through the origin that contains A and B.
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