MATH 110 - Fall 2002 - Ribet - Final (solution)

# MATH 110 - Fall 2002 - Ribet - Final (solution) - Math 110...

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C Math 110 PROFESSOR KENNETH A. RIBET Final Exam December 12, 2002 12:30–3:30 PM The scalar ﬁeld F will be the ﬁeld of real numbers unless otherwise speciﬁed. Please put away all books, calculators, electronic games, cell phones, pagers, .mp3 players, PDAs, and other electronic devices. You may refer to a single 2-sided sheet of notes. Please write your name on each sheet of paper that you turn in. Don’t trust staples to keep your papers together. Explain your answers as is customary and appropriate. Your paper is your ambassador when it is graded. Disclaimer: These solutions were written by Ken Ribet. As usual, sorry if they’re a bit terse and apologies also if I messed something up. If you see an error, send me e-mail and I’ll post an updated document. 1. Let T : V V be a linear transformation. Suppose that all non-zero elements of V are eigenvectors for T . Show that T is a scalar multiple of the identity map, i.e., that there is a λ R such that T ( v ) = λv for all v V . We can and do assume that V is non-zero. Choose v non-zero in V and let λ be the eigenvalue for v . We must show that Tw = λw for all w W . This is clear if w is a multiple of v . If not, w and v are linearly independent, so that w + v is non-zero, in particular. In this case, let μ be the eigenvalue of w and let a be the eigenvalue of w + v . Then a ( w + v ) = T ( w + v ) = Tw + Tv = μw + λv , and so ( a - μ ) w = ( λ - a ) v . Because v and w are linearly independent, we get a = μ and a = λ . Hence μ = λ . 2.

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MATH 110 - Fall 2002 - Ribet - Final (solution) - Math 110...

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