mathdefns.220.06 - ECE 220 Multimedia Signal Processing...

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ECE 220 Multimedia Signal Processing August 29, 2006 Fall 2006 Vector Spaces, Inner Products, and the Continuous-Time Fourier Series We will put the CTFS in a mathematical framework by defining it in terms of vector spaces and inner products . Vector Space (see for more) A vector space V is a set that is closed under finite vector addition and scalar multiplication. For a general vector space, the scalars are members of a field F , in which case V is called a vector space over F . In order for V to be a vector space, the following conditions must hold for all elements and any scalars : 1. Commutativity 2. Associativity of vector addition 3. Additive identity (existence of a zero vector, 0). For all X , 4. Existence of additive inverse. For any X , there exists a vector - X such that . 5. Associativity of scalar multiplication 6. Distributivity of scalar sums . 7. Distributivity of vector sums . 8. Scalar multiplication identity (existence of scalar 1) At this point, you are most familiar with vector spaces in n -dimensional Euclidean space. For example, the vector space of length-2 vectors represents the plane, the vector space of length-3 vectors represents 3-D space, etc. Verify that the above 8 conditions hold for these two vector spaces over the field of real numbers.
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