Let n be an even positive integer. In both parts of this Problem let V be the subspace of all vectors x → in ℝ n such that x j + 2 = x j + x j + 1 , for all j = 1 , ... , n − 2 . (In Exercise 70 we consider the special case n = 4 .) Consider the basis v → , w → of V with a → = [ 1 a a 2 ⋮ a n − 1 ] , b → = [ 1 b b 2 ⋮ b n − 1 ] , where a → = 1 + 5 2 and b = 1 − 5 2 . (In Exercise 4.3.72 we consider the case n = 4 .) a. Show that a → is orthogonal to b → . b. Explain why the matrix P of the orthogonal projection onto V is a Hankel matrix. Sec Exercises 71 and 72.
Let n be an even positive integer. In both parts of this Problem let V be the subspace of all vectors x → in ℝ n such that x j + 2 = x j + x j + 1 , for all j = 1 , ... , n − 2 . (In Exercise 70 we consider the special case n = 4 .) Consider the basis v → , w → of V with a → = [ 1 a a 2 ⋮ a n − 1 ] , b → = [ 1 b b 2 ⋮ b n − 1 ] , where a → = 1 + 5 2 and b = 1 − 5 2 . (In Exercise 4.3.72 we consider the case n = 4 .) a. Show that a → is orthogonal to b → . b. Explain why the matrix P of the orthogonal projection onto V is a Hankel matrix. Sec Exercises 71 and 72.
Solution Summary: The author explains that the two vectors are orthogonal.
Let n be an even positive integer. In both parts of this Problem let V be the subspace of all vectors
x
→
in
ℝ
n
such that
x
j
+
2
=
x
j
+
x
j
+
1
, for all
j
=
1
,
...
,
n
−
2
. (In Exercise 70 we consider the special case
n
=
4
.) Consider the basis
v
→
,
w
→
of V with
a
→
=
[
1
a
a
2
⋮
a
n
−
1
]
,
b
→
=
[
1
b
b
2
⋮
b
n
−
1
]
, where
a
→
=
1
+
5
2
and
b
=
1
−
5
2
. (In Exercise 4.3.72 we consider the case
n
=
4
.) a. Show that
a
→
is orthogonal to
b
→
. b. Explain why the matrix P of the orthogonal projection onto V is a Hankel matrix. Sec Exercises 71 and 72.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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