u6 - Unit VI: Eigenvectors, Eigenvalues, and...

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Unformatted text preview: Unit VI: Eigenvectors, Eigenvalues, and Diagonalization 1. Matrix of a Linear Transformation with Respect to a Basis Recall that a linear transformation on R n is a function T from R n to R n that satisfies T ( a x + b y ) = aT ( x ) + bT ( y ) , x , y ∈ R n , a,b scalars . Associated with T is the n × n matrix A defined so that T ( x ) = A x for x ∈ R n . The k th column of A contains the coordinates of T ( e k ) with respect to the standard basis, T ( e k ) = n X j =1 A jk e j , 1 ≤ k ≤ n. We extend this definition to an arbitrary basis { v 1 ,..., v n } for R n by defining the matrix of T with respect to the basis { v 1 ,..., v n } to be the n × n matrix A satisfying T ( v k ) = n X j =1 A jk v j , 1 ≤ k ≤ n. Note that the summation is over the first index, not the second. Thus the k th column of A contains the coordinates of T ( v k ) with respect to the basis { v 1 ,..., v n } . The definition of a matrix with respect to a basis depends on the specific order of the vectors v 1 ,..., v n in the basis. With this in mind, we occasionally refer to { v 1 ,..., v n } as an ordered basis for R n . Example: Let T : R 2 → R 2 be the reflection in the line { y = x } . Since T ( e 1 ) = e 2 and T ( e 2 ) = e 1 , the matrix of T with respect to the standard basis { e 1 , e 2 } is A = 0 1 1 0 . The vectors w 1 = e 1 + e 2 and w 2 =- e 1 + e 2 also form a basis for R 2 . Since T ( w 1 ) = w 1 and T ( w 2 ) =- w 2 , the matrix of T with respect to the basis { w 1 , w 2 } is the diagonal matrix B = 1- 1 . Let { v 1 ,..., v n } be a basis for R n . Each v ∈ R n can be expressed uniquely as a linear combination v = ∑ c j v j , where the scalars c 1 ,...,c m are the coordinates of v with respect to the basis. Then T ( v ) = T ( ∑ k c k v k ) = ∑ k c k T ( v k ) = ∑ k c k ( ∑ j A jk v j ). If we interchange the order of summation, this latter sum becomes ∑ j ( ∑ k A jk c k ) v j . This shows that T ( v ) = ∑ d j v j , where d j = ∑ k A jk c k . Thus the coordinates d 1 ,...,d m of T ( v ) are related to the coordinates c 1 ,...,c m of v by the matrix equation d = A c . We state our result formally. Theorem 1. Let T be a linear transformation on R n , let { v 1 ,..., v n } be a basis for R n , and let A be the matrix of T with respect to { v 1 ,..., v n } . If v = ∑ n k =1 c k v k , then T ( v ) = ∑ n j =1 d j v j , where d j = n X k =1 A jk c k , 1 ≤ j ≤ n. 1 Note that the summation is over the second index, not the first. This apparent inconsis- tency with the formula for the images of the basis vectors has been the bane of linear algebra students for generations. As a well known mathematician Paul Halmos once pointed out, this inconsistency is not a perversity of mathematicians but rather a perversity of nature....
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This note was uploaded on 06/25/2008 for the course MATH 33a taught by Professor Lee during the Spring '08 term at UCLA.

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u6 - Unit VI: Eigenvectors, Eigenvalues, and...

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