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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 nonzero. 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.
 Spring '08
 Staff
 Linear Algebra, Algebra, Vectors

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