MIT18_024s11_ChAnotes

MIT18_024s11_ChAnotes - Linear Spaces we have seen...

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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) = + cB. 7. (Multiplication by unity) 1A = Definition. More generally, let V be any set of objects (which we call vectors). And suppose there are two operations on 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 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 see. Example 1. The subset of consisting of the 9-tuple alone is a subspace it ths "smallest possible" sub- Vn; space. Pad course V, by definition a Vn; "largest 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 = is a subspace of . 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 ---A/ We generalize the construction given in the preceding examples as follows: Definition. Let S = $A?, .
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MIT18_024s11_ChAnotes - Linear Spaces we have seen...

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