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29
CHALLENGE 5.16.
Form
y
=
U
∗
b
n
2
multiplications
Form
z
=
Σ
−
1
y
n
multiplications (
z
i
=
y
i
/σ
i
)
Form
x
=
Vz
n
2
multiplications
Total:
2
n
2
+
n
multiplications
CHALLENGE 5.17.
(a) The columns of
U
corresponding to nonzero singular values form such a basis,
since for any vector
y
,
Ay
=
U
Σ
V
∗
y
=
n
X
j
=1
u
j
(
Σ
V
∗
y
)
j
=
X
σ
j
>
0
u
j
(
Σ
V
∗
y
)
j
,
so any vector
Ay
can be expressed as a linear combination of these columns of
U
.
Conversely, any linear combination of these columns of
U
is in the range of
A
,so
they form a basis for exactly the range.
(b) Similar reasoning shows that the remaining columns of
U
form this basis.
CHALLENGE 5.18.
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This note was uploaded on 01/21/2012 for the course MAP 3302 taught by Professor Dr.robin during the Fall '11 term at University of Florida.
 Fall '11
 Dr.Robin

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