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Practice Midterm 2

Practice Midterm 2 - Math 225 Spring 2008 G Rosen Practice...

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Math 225 Spring 2008 G. Rosen Practice Problems for Second Midterm 1. Let T:V W be a linear transformation and let { v 1 , v 2 ,. . . , v n } be a basis for V. Show that T is 1 – 1 and onto if and only if {T( v 1 ), T( v 2 ), . . . , T( v n )} is a basis for W. 2. Let V be the set of all sequences {a i } i=1, = { a 1 , a 2 , a 3 , a 4 , . . . } of real numbers. a) Show that V is a linear vector space over the reals with addition and scalar multiplication defined in the usual way. b) Show that V must be infinite dimensional by showing that it contains an infinite set of linearly independent vectors. c) Let L: V V be the left shift operator defined by L({a i } i=1, ) = L({ a 1 , a 2 , a 3 , a 4 , . . . }) = { a 2 , a 3 , a 4 , . . . }, and show that L is a linear transformation of V into itself. d) Show that L is NOT 1-1, find the kernel of L, Ker(L), what is dim(Ker(L)), and find a basis for Ker(L). e) Show that L is onto V. 3. Given the matrix A 0 1 0 1 0 0 0 0 1 A = a. Find the eigenvalues of A. b. Find an orthonormal eigenbasis (with repect to the matrix A) for R 3 c. Find a similarity transform which diagonalizes A. (Helpful Time Saving Hint: If S is a matrix whose columns are orthonormal, then S -1 = S T .) d. Compute e
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