notes4-4 - SECTION 4.4 COORDINATE SYSTEMS Today the Santa...

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SECTION 4.4 COORDINATE SYSTEMS Today, the Santa Claus development begins to thin again. Suppose we have any vector space V with a basis B = { b 1 ,..., b p } . Pause a moment to ponder what this might mean. Now, suppose we have any vector x in V . The coordinates of x relative to the basis B (or the B -coordinates of x ) are weights c 1 ,c 2 ,...,c p such that x = c 1 b 1 + c 2 b 2 + ··· + c p b p . How do we know there even are such coordinates for every vector? Could a vector have two different lists of coordinates relative to the basis B ? If we list the B -coordinates of x in a column, we have a vector [ x ] B in R p called the coordinate vector of x (with respect to the basis B ). The function or mapping x [ x ] B is the coordinate mapping determined by B . It takes a vector in the vector space V , which may be polynomials or sequences or . . . , to a vector in R p . EXAMPLES. Let’s play with the list of vectors B = 1 - 1
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This note was uploaded on 05/07/2010 for the course M 56945 taught by Professor Danielallcock during the Spring '09 term at University of Texas.

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notes4-4 - SECTION 4.4 COORDINATE SYSTEMS Today the Santa...

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