C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_340L%20notes4-4

C_DOCUME~1_MAXWID~1 - B as columns then we get a matrix P B As in the example x = P B x B The matrix P B is called the change-of-coordinate matrix

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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 . The function or mapping x [ x ] B is the coordinate mapping determined by B . EXAMPLES. Let’s play with the list of vectors B = 1 - 1 1 , - 1 2 0 , - 2 3 1 . Is B a basis? For what?
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Suppose [ x ] B = 2 - 1 4 . Find x . Suppose x = 3 - 2 2 . Find x B .
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If we are working in R p and we use the vectors in a basis
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Unformatted text preview: B as columns, then we get a matrix P B . As in the example, x = P B [ x ] B . The matrix P B is called the change-of-coordinate matrix from B to the standard basis in R p . The matrix P B is invertible. WHY AND HOW CAN WE USE THIS FACT? THE THINNING FACT, FINALLY. For any vector space V with basis B , the coordi-nate mapping x → [ x ] B is a one-to-one linear transformation from V onto R p . The coordinate mapping makes any vector space with a finite basis look and act just like R p . This means we can use coordinates to test linear independence and spanning in any vector space. EXAMPLE. Use coordinate vectors to test the linear independence of the polynomials 1-2 t 2-3 t 3 , t + t 3 , 1 + 3 t-2 t 2 . HOMEWORK: SECTION 4.4...
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This note was uploaded on 04/15/2008 for the course M 340L taught by Professor Pavlovic during the Spring '08 term at University of Texas at Austin.

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C_DOCUME~1_MAXWID~1 - B as columns then we get a matrix P B As in the example x = P B x B The matrix P B is called the change-of-coordinate matrix

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