This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View Full
Document
 Spring '08
 All
 Linear Algebra, Algebra, Vector Space

Click to edit the document details