change-of-basis

# change-of-basis - CHAPTER 4 Vector spaces Section 4.5...

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Unformatted text preview: CHAPTER 4: Vector spaces Section 4.5: Change of Basis We have seen how to find different bases for a vector space V and how to find the coordinates of a vector vectorv ∈ V with respect to any basis B of V . In some cases, it is useful to have a quick way of determining the determining the coordinates of vectorv with respect to some basis C for V given the coordinates of vectorv with respect to the basis B . For example, in some applications it is useful to write polynomials in terms of powers of x- c . That is, given any polynomial p ( x ) = a + a 1 x + ··· + a n x n , you want to write it as p ( x ) = b + b 1 ( x- c ) + b 2 ( x- c ) 2 + ··· + b n ( x- c ) n Such a situation may arise if the values of x you are working with are very close to c . If you are working with many polynomials, then it would be very helpful to have a fast way of converting each polynomial. We can rephrase this problem in terms of linear algebra. Let S = { 1 , x, . . . , x n } be the standard basis for P n . Then, given [ p ( x )] S = a . . . a n we want to determine [ p ( x )] B where B is the basis B = { 1 , x- c, ( x- c ) 2 , . . . , ( x- c ) n } for P n . Observe that since taking coordinates is a linear operation, we get [ p ( x )] B = [ a 1 + a 1 x + ··· + a n x n ] B = a [1] B + a 1 [ x ] B + ··· + a n [ x n ] B = bracketleftbig [1] B [ x ] B ··· [ x n ] B bracketrightbig a . . . a n Thus, the multiplication of this matrix and the coordinate vector of p ( x ) with respect to the standard basis gives us the coordinates of p ( x ) with respect to the new basis B . We call this matrix the change of coordinates matrix from S coordinates to B coordinates and is denoted B P S ....
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change-of-basis - CHAPTER 4 Vector spaces Section 4.5...

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