# hw5 - are the corresponding eigenvectors? 3. Deﬁne T : R...

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HW # 5 Due date: 3/15/2011 1. Prove the “splitting lemma”: Suppose v 1 ,...,v n are linearly indepen- dent vectors. Show that span { v 1 ,...,v n } = span { v 1 ,...,v k }⊕ span { v k +1 ,...,v n } for all 1 k n . 2. Let T ∈ L ( V ) be a linear operator and v 1 ,...,v n be a basis of V such that M ( T ; ( v 1 ,...,v n )) = b 1 b 2 . . . b n Demonstrate how to ﬁnd the eigenvalues of this diagonal matrix. What
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Unformatted text preview: are the corresponding eigenvectors? 3. Deﬁne T : R 2 → R 2 by T ( x,y ) = (-y,x ). Show that T has no eigenvalues. and page 94: 2,5,6,7,10,11 Hint for # 11: Along the way to proving this you will show that if v is an eigenvector of ST , then Tv will be an eigenvector of TS as long as Tv is non-zero. 1...
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## This note was uploaded on 05/09/2011 for the course MAT 310 taught by Professor Staff during the Spring '08 term at SUNY Stony Brook.

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