MATH2107-4-40 - Section 4.4 Coordinate Systems Linear...

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Unformatted text preview: Section 4.4 Coordinate Systems Linear Algebra and its Applications David C. Lay -- Winter 2008 p. 1/ Given a basis B = {b1 , . . . , bn } for a vector space V . For each x V there exists a unique set of scalars c1 , c2 , . . . , cn such that x = c1 b1 + c2 b2 + + cn bn The scalars c1 , c2 , . . . , cn are called the coordinates of x relative to the basis B. These coordinates can be denoted by c1 . [x]B = . . cn -- Winter 2008 p. 2/ and called the coordinate vector of x relative to B or the B-coorindate vector of x. The mapping V Rn x [x]B is called the coordinate mapping. -- Winter 2008 p. 3/ Graphical Interpretation Suppose e1 = basis B for R2 . 1 0 , e2 = 0 1 are the vectors in the -- Winter 2008 p. 4/ Suppose b1 = e1 , b2 = 1 . Let B1 = {b1 , b2 }. 2 -- Winter 2008 p. 5/ Example 2 -1 Let b1 = , b2 = 1 1 B = {b1 , b2 }. Find [x]B . ,x = 4 . 5 -- Winter 2008 p. 6/ PB is called the change of coordinates matrix from B to the standard basis in R4 . PB is invertible by the Invertible Matrix Theorem. -1 PB x = [x]B . -1 From here PB represents the coordinate mapping x [x]B . -1 PB Rn - Rn : x [x]B -- Winter 2008 p. 7/ Theorem Let B = {b1 , . . . , bn } be a basis for a vector space V . Then the coordinate mapping x [x]B is a 1-1 linear transformation from V to Rn . -- Winter 2008 p. 8/ ...
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This note was uploaded on 05/26/2010 for the course MATH Math2107 taught by Professor Lanihaque during the Fall '10 term at Carleton.

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MATH2107-4-40 - Section 4.4 Coordinate Systems Linear...

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