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CHAPTER 6: INNER PRODUCT SPACES
Inner Products 1.
(a)
ǡ
ͳሻʹሺͲሻൌ
(b)
ሺ͵ǡʹሻൌͳͷͳͲൌʹͷ
(c)
ǡ
ͳǡͲሻൌ
(d)
ԡܝԡ ൌ ξ
!^
ൌ ξʹ
(a)
ǡ
(b)
(c)
ǡ
(d)
ൌ^
ͷ
(e)
ሺܝǡ ܞሻ ൌ ԡ ܝെ ܞԡ ൌ ۃሺെʹǡ െͳሻǡ ሺെʹǡ െͳሻۄ݀
ଵȀଶ
ൌ ሾʹሺെʹሻሺെʹሻ ͵ሺെͳሻሺെͳሻሿ
ଵȀଶ
ൌ ξͳͳ
(f)
ԡ ܝെ ܞ ݇ԡ ൌ ۃሺെͺǡ െͷሻǡ ሺെͺǡ െͷሻۄ
ଵȀଶ
ൌ ሾʹሺെͺሻሺെͺሻ ͵ሺെͷሻሺെͷሻሿ
ଵȀଶ
ൌ ξʹͲ͵
(a)
ܝۃǡ ۄܞൌ Ͷሺ͵ሻሺͶሻ ͷሺെʹሻሺͷሻ ൌ െʹ
;^
ܞۃǡ ۄܝൌ ͶሺͶሻሺ͵ሻ ͷሺͷሻሺെʹሻ ൌ െʹ
(b)
ܝۃ ܞǡ ۄܟൌ Ͷሺሻሺെͳሻ ͷሺ͵ሻሺሻ ൌ ʹ
ܝۃǡ ۄܟ ܞۃǡ ۄܟൌ Ͷሺ͵ሻሺെͳሻ ͷሺെʹሻሺሻ ͶሺͶሻሺെͳሻ ͷሺͷሻሺሻ ൌ ʹ (c)
ܝۃǡ ܞ ۄܟൌ Ͷሺ͵ሻሺ͵ሻ ͷሺെʹሻሺͳͳሻ ൌ െͶ
ܝۃǡ ۄܞ ܝۃǡ ۄܟൌ Ͷሺ͵ሻሺͶሻ ͷሺെʹሻሺͷሻ Ͷሺ͵ሻሺെͳሻ ͷሺെʹሻሺሻ ൌ െͶ (d)
ܝ݇ۃǡ ۄ ܞൌ ͶሺെͳʹሻሺͶሻ ͷሺͺሻሺͷሻ ൌ ͺ
;݇^
ܝۃǡ ۄܞൌ െͶሾͶሺ͵ሻሺͶሻ ͷሺെʹሻሺͷሻሿ ൌ ሺെͶሻሺെʹሻ
;^ ܝۃǡ ۄ ܞ ݇ൌ Ͷሺ͵ሻሺെͳሻ ͷሺെʹሻሺെʹͲሻ ൌ ͺ
(e)
ۃǡ ۄܞൌ ͶሺͲሻሺͶሻ ͷሺͲሻሺͷሻ ൌ Ͳ
;^ ܞۃǡ ۄൌ ͶሺͶሻሺͲሻ ͷሺͷሻሺͲሻ ൌ Ͳ
Chapter 6: Inner Product Spaces
5.^
(c)
^
§^
·^
§^
^
^
¨^
¸^
¨^
©^
¹^
©^
ʹ^
ͳ^
ʹ^
ʹ^
ͳ^
ͳ^
ͷ^
൏^
ǡ^
ൌ^
ൌ^
ͳ͵
ͳ^
ͳ^
ͳ^
ͳ^
ͳ^
Ͳ^
͵^
ͳ
(d)
ൌ^
ͷǡ ͵ Ǥ ͷǡ ͵ ൌ
6.^
(a)
ǡ
^
^
^
^
§^
·^
§^
^
^
¨^
¸^
¨^
©^
¹^
©^
Ͳ^
ͳ^
ʹ^
Ͳ^
ͳ^
ͳ^
ͳ^
ͳ
ൌ^
ൌ^
ʹ^
ͳ^
ͳ^
ʹ^
ͳ^
ͳ^
ͷ^
ͳ
(b)
ǡ
^
^
^
§^
·^
§^
^
¨^
¸^
¨^
^
^
©^
¹^
©^
Ͳ^
ͳ^
ͳ^
Ͳ^
ͳ^
Ͳ^
ͳ^
ͳ
ൌ^
ʹ^
ͳ^
ͳ^
ʹ^
ͳ^
ͳ^
ͳ^
ͳ
(c)
^
^
^
§^
·^
§^
^
^
¨^
¸^
¨^
©^
¹^
©^
Ͳ^
ͳ^
ʹ^
Ͳ^
ͳ^
ͳ^
ͳ^
ൌ^
ͳͲ
ʹ^
ͳ^
ͳ^
ʹ^
ͳ^
Ͳ^
ͷ^
(d)
^
ൌ^
ͳǡ ͷ
ͳǡ ͷ ൌ
(e)
݀ ሺܞǡ ܟሻ ൌ ԡ ܞെ ܟԡ ൌ ቂቀቂ
ͳ^
ʹ^
െͳ
െͳʹ
ቃቁ ή ቀቂ
ͳ^
ʹ^
െͳ
െͳʹ
ଵȀଶ
െͳെͶ
ቃ ή ቂ
െͳെͶ
ൌ ξͳ
(f)
ԡ ܞെ ܟԡ
ଶ^ ൌ ቀቂ
ͳ^
ʹ^
െͳ
െͳʹ
ቃቁ ή ቀቂ
ͳ^
ʹ^
െͳ
െͳʹ
െͳെͶ
ቃ ή ቂ
െͳെͶ
ቃ ൌ ͳ
8.^
(a)
ܙ ǡܘ
ǡʹ
(b)
ܘۃǡ ۄܙൌ ሺെͷሻሺ͵ሻ ሺʹሻሺʹሻ ሺͳሻሺെͶሻ ൌ െͳͷ
10.
(a)
^
^
§^
·^
§^
·^
§^
§^
·^
§^
^
¨^
¸^
¨^
¸^
¨^
¨^
¸^
¨^
¸^
^
^
©^
¹^
©^
©^
¹^
©^
¹^
©^
^
^
^
^
^
^
^
^
ͳ^
ͳ^
ͳ^
ʹ^
ͳ^
ʹ
ʹ^
ʹ^
ͳ^
ʹ^
ͳ^
ʹ
͵^
ʹ^
͵^
͵^
ʹ^
͵^
൏^
ǡ^
^
ͳ^
ͳ^
ͳ^
ͳ
^
^
^
^
^
^
^
^
^
^
^
^
ͳ^
ʹ^
ͳ^
ʹ^
ͳ^
ʹ^
ͳ^
ʹ^
ͳ^
ͳ^
ʹ^
ͳ^
ͳ^
ʹ^
ʹ^
ʹ
ʹ^
͵^
ʹ^
^
ൌͳͲ
^
^
ͷ
(b)
ܝۃǡ ۄܞൌ ͷሺͲሻሺሻ െ ሺͲሻሺʹሻ െ ሺെ͵ሻሺሻ ͳͲሺെ͵ሻሺʹሻ ൌ െͶʹ
(a)
ۄܘ ǡܘۃ ൌ ԡܘԡ
ଵȀଶ
ଶ^ Ͷ
ଶ^ ሺെ͵ሻ
ଶ^ ൌ ξʹͻ
(b)
ۄܘ ǡܘۃ ൌ ԡܘԡ
ଵȀଶ
ଶ^ Ͳ
ଶ^ Ͷ
ଶ^ ൌ ͷ
13. (a)
§^
¨^
©^
ʹͻ
ൌ^
^
ൌ^
^
ൌ^
͵^
Ͷͷ
Chapter 6: Inner Product Spaces
ǡ^
Ͳ ʹ ͳ
ͳ ͵ Ͷ ൌ
ͳͳ
^
The inner product is the evaluation inner product at
െʹǡ Ͳǡ
and 2.
(a)
ܘۃǡ ۄܙൌ ሺͳ Ͷ ͳʹሻሺ͵ Ͷሻ ሺͳሻሺ͵ሻ ሺͳ െ Ͷ ͳʹሻሺ͵ Ͷሻ ൌ ͳͺͷ (b)
ۄܘ ǡܘۃ ൌ ԡܘԡ
ଵȀଶ
ൌ ඥሺͳ Ͷ ͳʹሻሺͳ Ͷ ͳʹሻ ሺͳሻሺͳሻ ሺͳ െ Ͷ ͳʹሻሺͳ െ Ͷ ͳʹሻ
ൌ ͵ͳ (c)
ሺܘǡ ܙሻ ൌ ԡ ܘെ ܙԡ ൌ ԡെʹ െ ʹ ݔ ʹݔ݀
ଶ^ ԡ ൌ ඥ
ൌ ͳͲͺ
ܙ ǡܘ
ͳሻ
ͳሻ
^
^
ൌͳͲሺͷሻሺ
ʹሻሺʹሻͲሺͳሻʹሺʹሻൌͷͲ
^
ʹ^
ʹ^
ʹ^
ʹ^
ʹ
ʹ^
ʹ
ൌ^
ʹ^
^
ͳ^
Ͳ^
^
ͳ^
ൌ^
ͳͲ
ʹ^
ͳͲͺ
(a)
Using the Euclidean inner product,
ۄܟ ǡܟۃ ൌ ԡܟԡ
ଵȀଶ
ଶ^ ሺെͷሻ
ଶ^ ൌ ξʹͻ
(b)
Using the weighted Euclidean inner product,
ۄܟ ǡܟۃ ൌ ԡܟԡ
ଵȀଶ
ൌ ඥʹሺʹሻሺʹሻ ͵ሺെͷሻሺെͷሻ ൌ ξͺ͵
(c)
Using the matrix inner product associated with
, note first that
െͳ
ʹ^
ʹ െͷ
െʹʹെͳͳ
Then
ۄܟ ǡܟۃԡܟԡ
ଵȀଶ
ଵȀଶ
ൌ ඥሺെʹʹሻሺെʹʹሻ ሺെͳͳሻሺെͳͳሻ ൌ ͳͳξͷ
(a)
ۄܟ ǡܞۃ ۄܞ ǡܞۃ ۄܟ ǡܝۃ ۄܞ ǡܝۃ ൌ ۄܟ ܞ ǡܞۃ ۄܟ ܞ ǡܝۃ ൌ ۄܟ ܞ ǡܞ ܝۃ ԡܞԡ ۄܟ ǡܝۃ ۄܞ ǡܝۃ ൌ
ଶ^ ܞۃǡ ۄܟൌ ʹ ͷ Ͷ െ ͵ ൌ ͺ
(b)
ۄܟʹ ܝ͵ ǡܟۃ െ ۄܟʹ ܝ͵ ǡܞʹۃൌ ۄܟʹ ܝ͵ ǡܟെ ܞʹۃ
^
ۄܟʹ ǡܟۃ െ ۄܝ͵ ǡܟۃ െ ۄܟʹ ǡܞʹۃ ۄܝ͵ ǡܞʹۃൌ ԡܟെ ʹԡ ۄܟ ǡܝۃ͵ െ ۄܟ ǡܞۃͶ ۄܞ ǡܝۃ ൌ ۄܟ ǡܟۃʹ െ ۄܝ ǡܟۃ͵ െ ۄܟ ǡܞۃͶ ۄܝ ǡܞۃ ൌ
ଶ^
ൌ ሺʹሻ Ͷሺെ͵ሻ െ ͵ሺͷሻ െ ʹሺͶͻሻ ൌ െͳͳ͵
Chapter 6: Inner Product Spaces
(c)
ۄܞ ܝͶ ǡܟʹۃെ ۄܞ ܝͶ ǡܞۃ െ ۄܞ ܝͶ ǡܝۃ ൌ ۄܞ ܝͶ ǡܟʹ െ ܞ െ ܝۃ
ۄܞ ǡܟʹۃെ ۄܝͶ ǡܟʹۃെ ۄܞ ǡܞۃ െ ۄܝͶ ǡܞۃ െ ۄܞ ǡܝۃ ۄܝͶ ǡܝۃ ൌ
ۄܞ ǡܟۃʹ െ ۄܝ ǡܟۃͺ െ ۄܞ ǡܞۃ െ ۄܝ ǡܞۃͶ െ ۄܞ ǡܝۃ ۄܝ ǡܝۃͶ ൌ ൌ Ͷԡܝԡ
ଶ^ ԡܞԡ െ ۄܞ ǡܝۃ͵ െ
ଶ^ െ ͺܝۃǡ ۄܟെ ʹܞۃǡ ۄܟൌ Ͷ െ ͵ሺʹሻ െ Ͷ െ ͺሺͷሻ െ ʹሺെ͵ሻ ൌ െͶͲ
(d)
ۄܞ ǡܞۃ ۄܝ ǡܞۃ ۄܞ ǡܝۃ ۄܝ ǡܝۃඥ ൌ ۄܞ ܝ ǡܞۃ ۄܞ ܝ ǡܝۃඥ ൌ ۄܞ ܝ ǡܞ ܝۃඥ ൌ ԡܞ ܝԡ^ ൌ ඥԡܝԡ
ଶ^ ԡܞԡ ۄܞ ǡܝۃʹ
ଶ^ ൌ ඥͳ ʹሺʹሻ Ͷ ൌ ͵
(e)
ۄܞ െ ܟʹ ǡܞۃ െ ۄܞ െ ܟʹ ǡܟʹۃඥ ൌ ۄܞ െ ܟʹ ǡܞെ ܟʹۃඥ ൌ ԡܞെ ܟԡʹ^ ԡܟൌ ඥͶԡ ۄܞ ǡܞۃ ۄܟʹ ǡܞۃ െ ۄܞ ǡܟʹۃെ ۄܟʹ ǡܟʹۃඥ ൌ
ଶ^ ԡܞԡ ۄܟ ǡܞۃͶ െ
ଶ
ൌ ඥͶሺͶͻሻ െ Ͷሺെ͵ሻ Ͷ ൌ ξʹͳʹ ൌ ʹξͷ͵ (f)
ԡ ܝെ ʹ ܞ Ͷܟԡ ൌ ඥ ܝۃെ ʹ ܞ Ͷܟǡ ܝെ ʹ ܞ Ͷۄܟ ۄܟͶ ܞʹ െ ܝ ǡܟͶۃ ۄܟͶ ܞʹ െ ܝ ǡܞʹۃെ ۄܟͶ ܞʹ െ ܝ ǡܝۃඥ ൌ ۄܟͶ ǡܟͶۃ ۄܞʹ ǡܟͶۃെ ۄܝ ǡܟͶۃ ۄܟͶ ǡܞʹۃെ ۄܞʹ ǡܞʹۃ ۄܝ ǡܞʹۃെ ۄܟͶ ǡܝۃ ۄܞʹ ǡܝۃ െ ۄܝ ǡܝۃඥ ൌ
ൌ ඥԡܝԡ
ଶ^ ԡܞ Ͷԡ ۄܟ ǡܝۃͺ ۄܞ ǡܝۃͶ െ
ଶ^ െ ͳܞۃǡ ۄܟ ͳԡܟԡ
ଶ
ൌ ඥͳ െ Ͷሺʹሻ ͺሺͷሻ ͶሺͶሻ െ ͳሺെ͵ሻ ͳሺͶͻሻ ൌ ξͺͺͳ
ൌ ۄܞ ǡܝۃ
ଵ ଽ^
ݒଶ^
ଶ^
(see Example 3 in Section 6.1)
(a)
Axiom 4 does not hold, e.g.,
ۃሺͲǡ ͳǡ Ͳሻǡ ሺͲǡ ͳǡ Ͳሻ ۄൌ Ͳ
; this is not an inner product on
(b)
Axiom 2 does not hold, e.g., with
ܝൌ ܞൌ ܟൌ ሺͳǡ Ͳǡ Ͳሻ
we have
Ͷ ൌ ۄܟ ǡܞ ܝۃ
but
ܝۃǡ ۄܟ ܞۃǡ ۄܟൌ ͳ ͳ ൌ ʹ
Axiom 3 does not hold either, e.g., with
ܝൌ ܞൌ ሺͳǡ Ͳǡ Ͳሻ
and
Ͷ ൌ ۄ ܞ ǡܝ݇ۃ
does not
equal
ʹ ൌ ۄܞ ǡܝۃ݇
; this is not an inner product on
(c)
This is a weighted Euclidean inner product (see Formula (1) of Section 6.1), which satisfies allfour axioms (d)
Axiom 4 does not hold, e.g.,
ۃሺͲǡ ͳǡ Ͳሻǡ ሺͲǡ ͳǡ Ͳሻ ۄൌ െͳ ൏ Ͳ
; this is not an inner product on
(a)
For
ܘൌ ͳ
,^ ۄܘ ǡܘۃ ൌ ԡܘԡ
ଵȀଶ
ݔ݀ͳ ଵିଵ^
ଵ^ ଵ
ൌ ξʹ
For
ۄܘ ǡܘۃ ൌ ԡܘԡ,
ଵȀଶ
ଵିଵ^
య௫ (^) ଷ^
ଶ ଷ
Chapter 6: Inner Product Spaces
The general solution of the homogeneous system
is therefore
ଷ ݐସ^
,^ ݔ
ଵଷ ଵସ
ଵ ଵଶ
and thus all solution vectors are scalar multiples of
ሺെͻǡ ͳ͵ǡ ͳǡ ͳʹሻ
. All
solution vectors of the homogeneous system are orthogonal to every row vector of thecoefficient matrix (see Example 6 of Section 6.2).To find unit vectors, we must normalize the solution above. We have ԡሺെͻǡ ͳ͵ǡ ͳǡ ͳʹሻԡ ൌ ඥሺെͻሻ
ଶ^ ͳ͵
ଶ^ ͳ
ଶ^ ͳʹ
ଶ^ ൌ ξ͵ͻͷ
so that two unit vectors that are orthogonal to all three of the given vectors are^ ଵ ξଷଽହ
ሺെͻǡ ͳ͵ǡ ͳǡ ͳʹሻ
and
ଵ ξଷଽହ
ሺെͻǡ ͳ͵ǡ ͳǡ ͳʹሻ
(a)
ȁܝۃǡ ۄܞȁ ൌ ȁ͵ሺെʹሻሺͳሻ ʹሺͳሻሺͲሻȁ ൌ ȁെȁ ൌ
ԡܝԡ ൌ ඥܝۃǡ ۄܝൌ ඥ͵ሺെʹሻሺെʹሻ ʹሺͳሻሺͳሻ ൌ ξͳͶ
ԡܞԡ ൌ ඥܞۃǡ ۄܞൌ ඥ͵ሺͳሻሺͳሻ ʹሺͲሻሺͲሻ ൌ ξ͵
since
ԡܝԡԡܞԡ ൌ ξͶʹ ξ͵ ൌ ൌ ȁܝۃǡ ۄܞȁ
, we conclude that the Cauchy-Schwarz
inequality holds (b)
ܸǡܷۃȁ
ۄȁ ൌ ȁሺെͳሻሺͳሻ ሺʹሻሺͲሻ ሺሻሺ͵ሻ ሺͳሻሺ͵ሻȁ ൌ ȁʹͲȁ ൌ ʹͲ
ܷԡ^
ܷǡܷۃඥ ൌ ԡ
ۄൌ ඥሺെͳሻሺെͳሻ ሺʹሻሺʹሻ ሺሻሺሻ ሺͳሻሺͳሻ ൌ ξͶʹ
ܸԡ^
ܸǡܸۃඥ ൌ ԡ
ۄൌ ඥሺͳሻሺͳሻ ሺͲሻሺͲሻ ሺ͵ሻሺ͵ሻ ሺ͵ሻሺ͵ሻ ൌ ξͳͻ
since
ܷԡ^
ܸԡԡ
ԡ ൌ ξͻͺ ξͶͲͲ ൌ ʹͲ ൌ ȁܷۃǡܸ
ȁۄ, we conclude that the Cauchy-Schwarz
inequality holds (c)
ȁܘۃǡ ۄܙȁ ൌ ȁሺെͳሻሺʹሻ ሺʹሻሺͲሻ ሺͳሻሺെͶሻȁ ൌ ȁെȁ ൌ
ԡܘԡ ൌ ඥܘۃǡ ۄܘൌ ඥሺെͳሻሺെͳሻ ሺʹሻሺʹሻ ሺͳሻሺͳሻ ൌ ξ
ԡܙԡ ൌ ඥܙۃǡ ۄܙൌ ඥሺʹሻሺʹሻ ሺͲሻሺͲሻ ሺെͶሻሺെͶሻ ൌ ξʹͲ
since
ԡܘԡԡܙԡ ൌ ξͳʹͲ ξ͵ ൌ ൌ ȁܘۃǡ ۄܙȁ
, we conclude that the Cauchy-Schwarz
inequality holds
If
^ Ȁ͵
is the subspace
, then
A^ is the line
47. (a)
If
is the plane given by equation
, then the vectors
ሺʹǡͲǡͳሻ
and
ሺͳǡʹǡͲሻ
are in
, so we know a basis for
A^ will be given by a vector orthogonal to both of these
vectors. We know the cross product will give us such a vector, so
§^
¨^
¸^
¨^
©^
ʹ^
Ͳ^
ͳ^
ʹǡ ͳǡ Ͷ
ͳ^
ʹ^
^
^
is a
basis for
A.
Chapter 6: Inner Product Spaces
(Alternatively, if the equation of the plane for
is
, we know the normal
vector is
ሺʹǡ
ͳǡ
(c)
The intersection of the planes
and
is the solution to the system
represented by the augmented matrix
^
§^
¨^
©^
ͳ^
ͳ^
ͳ^
ͳ^
ʹ^
ͳ^
Ͳ^
This row reduces to
(^1) 2
§^
¨^
¨^
©^
͵ ͵
ͳ^
Ͳ^
Ͳ^
ͳ^
so the interesection,
, is the line with basis given by
§^
¨^
¨^
©^
ͳ ʹ ͵¹
So,
A^ is a plane perpendicular to
this vector, hence the equation of the plane is
48. (a)
Let
§^
¨^
^
¨^
¨^
©^
ʹ^
ͳ^
ൌ^
ͳ^
Ͷ^
Ͷ^
ͷ^
ͳ͵
^
The nullspace of
^
is the set of all vectors perpendicular to all three
rows, so a basis for the nullspace will be a basis for the orthogonal complement of thesubspace generated by
^ ͳ
ǡ
, andʹ
. Since
^
row reduces to
§^
¨^
¨^
¨^
©^
ͳ^
Ͳ^
Ͳ^
ͳ^
ͳ
Ͳ^
Ͳ^
we have a basis
for the orthogonal complement given by
¨^
¨^
¨^
©^
ʹ^ .ͳ ͳ¹
(b)
Let
§^
¨^
¨^
¨^
©^
Ͳ^
ʹ^
ͳ
ൌ^
Ͷ^
Ͳ^
^
ͳ^
Then
^
row reduces to
(^3412)
§^
¨^
¨^
¨^
©^
ͳ^
Ͳ^
ͳ Ͳ^
Ͳ^
, so a basis for the orthogonal
complement is given by
§^
¨^
¨^
©^
(c)
Let
§^
¨^
^
^
¨^
¨^
©^
͵^
Ͳ^
ͳ^
ൌ^
ͳ^
ʹ^
ʹ^
ͳ
Ͷ^
ʹ^
͵^
Then
^
row reduces to
1
2 3
3 5
1 6
^6
§^
¨^
¨^
¨^
©^
ͳ^
Ͳ^
ͳ Ͳ^
Ͳ^
Ͳ^
so a basis for the ,
orthogonal complement is given by the vectors
¨^
¨^
¨^
¨^
©^
ʹ ͷ^ and Ͳ¹
§^
¨^
¨^
¨^
¨^
¨^
©^
Ͷ ͳ Ͳ ¹
Chapter 6: Inner Product Spaces
(d)
Begin by forming a matrix
whose rows are the given vectors:
ͳ^
Ͷ^
ͷ^
^
ͻ
͵^
ͳ^
Ͷ^
െͳ
െͳ
Ͳ^
െͳ
െͳ
ʹ^
͵^
ͷ^
^
has the reduced row echelon form
ͳ ൦
Ͳ^
ͳ^
ʹ^
ͳ
Ͳ^
ͳ^
ͳ^
ͳ^
Ͳ^
Ͳ^
Ͳ^
Ͳ^
Ͳ^
Ͳ^
Ͳ^
Ͳ^
. The
general solution of the homogeneous system
is
,^ ݔ
,^ ݔ
,^ ݔ
,^ ݔ
, therefore
ܠൌ ݎሺെͳǡ െͳǡ ͳǡ Ͳǡ Ͳሻ ݏሺെʹǡ െͳǡ Ͳǡ ͳǡ Ͳሻ
ݐሺെͳǡ െʹǡ Ͳǡ Ͳǡ ͳሻ
A basis for the orthogonal complement is formed by vectors
ሺെͳǡ െͳǡ ͳǡ Ͳǡ Ͳሻ
ሺെʹǡ െͳǡ Ͳǡ ͳǡ Ͳሻ
, and
ሺെͳǡ െʹǡ Ͳǡ Ͳǡ ͳሻ
Using the identity
ଵ ଶ^
we obtain
ǡ
ͳቆන ሺሺ ݇െ ݈ሻݔʹ
గ ݀
గ ݀
ݔ^
Substitute
in the first equation, and
ݐ ൌ ሺ݇ κሻݔ
in the second, so that
and
we get
ǡ
ͳቆʹ
since both
and
are integers and
for
an integer.
Gram-Schmidt Process; QR-Decomposition 65.
(a)
ሺͶǡͲሻ
ሺͲǡ
, so the vectors are orthogonal.
(b)
^
ͳȀ
ʹǡ
ͳȀ
ͳȀ
ʹǡ ͳȀ
, so the vectors are orthogonal.
(c)
ሺʹǡʹሻ
ͳȀʹǡ
ͳȀʹሻൌ
, so the vectors are not orthogonal.
(d)
͵ǡͳሻ
ሺͲǡͲሻൌͲ
, so the vectors are orthogonal.
66. (a)
Neither vector is a unit vector so they are not orthonormal. (b)
Both vectors have length 1, so they are orthonormal. (c)
Not orthonormal. (d)
Not orthonormal.
Chapter 6: Inner Product Spaces
67. (a)
If
ൌ ͲǡͳȀ
ʹǡ
ͳȀ
^
ͳ Ȁ
͵ǡ ͳȀ
͵ǡ ͳȀ
͵
and
ܟൌ ሺͲǡ ͳȀξʹǡ ͳȀξʹሻǡ
then
zͲ
, so the vectors are not orthogonal.
(c)
ሺͳǡͲǡͳሻ
^
^
^
z
ͳȀ ͵ǡ ͳȀ
͵ǡ
ͳȀ ͵
Ͳǡ^
so the vectors are not orthogonal.
(a)
Not orthonormal. (b)
The set is orthonormal (it is orthogonal, and the norm of each vector is 1) (c)
Not orthonormal. (d)
The set is orthonormal (it is orthogonal, and the norm of each vector is 1)
69. (a)
Let
1
2
2
2
1
2
2
2
1
^
ʹ^
ʹ^
ʹ
ͳ^
ʹ^
͵
͵^
͵^
͵^
͵^
͵^
͵^
͵^
͵^
͵
ൌ^
^
ǡ^
ൌ^
ǡ^
ൌ^
^
^
^
^
^
^
^
. Then
2
4
2
^
z
ͳ^
͵^
ͻ^
ͻ^
ͻ
^
^
Ͳǡ
^
^
so the vectors are not orthogonal, and hence not orthonormal.
70. (a)
Let
§^
¨^
©^
Ͳ^
Ͳ^
ͳ
1
2 ^2 §^
¨^
¨^
©^
͵^
͵ ͵ ൌ
^
2
1 ^2 §^
¨^
¨^
©^
͵^
͵ ͵ ൌ
^
and ,
2
2 §^1
¨^
¨^
©^
^
͵^
͵ ͵ ൌ^
Ͳ^
Verify that all the inner products are
, so the set of matrices are orthogonal. For example,
2
1 4
2
§^
^
¨^
¨^
ͻ^
ͻ ͻ^
ͻ
൏^
ǡ^
Also, note that the norms of all the matrices are 1. For
example
8
2 2
1
§^
¨^
¨^
ͻ^
ͻ ͻ^
ͻ
൏^
ǡ^
ൌͳ
,^ so
ൌͳ ^
. After all verifications, we see that this
is an orthonormal set of matrices. (b)
The norms of the last two matrices are
ξʹ് ͳ
therefore this is not an orthonormal set.
71. (a)
ሺʹǡ͵ሻ
ǡͶሻൌͲ
. Normalizing the orthonormal set is
^
`
2
3
6
4
ͳ͵^
ͳ͵^
ͷʹ^
ͷʹ
ǡ^
ǡ^
ǡ
(b)
Verify the vectors are orthogonal. The orthnormal set is
^
`
1
1
1
1
^
ʹ^
ʹ
ʹ^
ʹ ǡ^
ǡ Ͳ ǡ
ǡ^
ǡ Ͳ ǡ Ͳǡ Ͳǡͳ
Write
ܝൌ ሺͳǡ Ͳሻ
and
ܞൌ ሺͲǡ ͳ
). Then:
ܝۃǡ ۄܞൌ ͵ሺͳሻሺͲሻ ʹሺͲሻሺͳሻ ൌ Ͳ
so that the vectors are orthogonal. Thus we must normalize each of these vectors to get anorthonormal set
Chapter 6: Inner Product Spaces
(a)
ܠൌ ቀͳǡௐ
ଵଵ
ǡ െ
ଵ ǡଷ^
ଵ ቁ^
has been calculated in the solution of Exercise 80(a) above.
ൌ ሺͳǡ ʹǡ Ͳǡ െͳሻ െ ቀͳǡଵ
ଵଵ
ǡ െ
ଵ ǡଷ^
ଵ ቁ ൌ ቀͲǡ^
ଵ ^
ଵǡ ଷ^ ǡ െ
ቁ^
(b)
ܞ ǡܠۃ ൌ ܠௐ
ܞ ǡܠۃ ଵ
ܞ ǡܠۃ ଶ
ଷ
ൌ^
ାଶାାଵ
ξଵ଼
ቀͲǡ
ଵ ξଵ଼
ǡ െ
ସ ξଵ଼
ǡ െ
ଵ ξଵ଼
ଵଶ^
ହ ଷ^
ଵ ቁ ቀ^
ଵǡଶ^
ହ ǡ^
ଵ ^
ଵǡ ቁ ^
ଵାାାସ
ξଵ଼
ቀ^
ଵ ξଵ଼^
ǡ Ͳǡ
ଵ ξଵ଼
ǡ െ
ସ ξଵ଼
ൌ ቀͲǡ
ଵ ǡ െ^
ଶ ǡ െଷ^
ଵ ቁ ቀͳǡ^
ହ ǡଷ^
ଵ ǡଷ^
ଵ ቁ ቀଷ^
ହ ଵ଼^ ǡ Ͳǡ
ହ ଵ଼^ ǡ െ
ଵ ଽ
ଶଷଵ଼^
ଵଵǡ
ǡ െ
ଵ ଵ଼^ ǡ െ
ଵ ଵ଼^
ൌ ሺͳǡ ʹǡ Ͳǡ െͳሻ െ ቀଵ
ଶଷଵ଼^
ଵଵǡ
ǡ െ
ଵ ଵ଼^ ǡ െ
ଵ ଵ଼^
ହ ଵ଼^
ଵǡ ^
ଵǡ ଵ଼^
ǡ െ
ଵ ଵ଼^
(a)
First use Gram-Schmidt to get an orthogonal basis:^ ܞ
ൌ ሺͳǡ ͳǡ ͳሻଵ
ܝۃ ܞǡమ^
ۄభ ܞԡభ
మ^ ԡ
^
ൌ ሺͳǡ Ͳǡ െͳሻ െ
ଵାାሺିଵ
ሻ
ା
^ ା
ሺͳǡ ͳǡ ͳሻ
^
ൌ ሺͳǡ Ͳǡ െͳሻ െ Ͳሺͳǡ ͳǡ ͳሻ ൌ ሺͳǡ Ͳǡ െͳሻ ܞଷ
ܝۃ
ܞǡǤయ
ۄభ ܞԡభ
మ^ ԡ
ܝۃ
ܞǡǤయ
ۄమ ܞԡమ
^ ԡ
ൌ ሺʹǡ ͳǡ െͳሻ െ
ଶାଵାሺିଵ
ሻ
^ ା
^ ା
^
ሺͳǡ ͳǡ ͳሻ െ
ଶାାଵ ^ ା
^ ାሺିሻ
ሺͳǡ Ͳǡ െͳሻ
ൌ ሺʹǡ ͳǡ െͳሻ െ
ଶሺͳǡ ͳǡ ͳሻ െ ଷ^
ଷሺͳǡ Ͳǡ െͳሻ ଶ^
ଵ ǡ^
ଵ ଷ ǡ െ
ଵ ቁ^
Then an orthonormal basis is
ܞ భܞԡԡభ^
ଵ ξଷ
ሺͳǡ ͳǡ ͳሻ ൌ ቀ
ଵ ξଷ
ǡ^ ଵǡ^ ξଷ
ଵ ξଷ
ܙ^ ଶ
ܞ మܞԡԡ^ మ^
ൌ^
ଵ ξଶ^
ሺͳǡ Ͳǡ െͳሻ ൌ ቀെ
ଵ ξଶ
ǡ Ͳǡ െ
ଵ ξଶ
ܙ^ ଷ
ܞ ܞԡԡ^ ^
ൌ^
ଵ ଵȀξ
ଵ ǡ^
ଵ ଷ ǡ െ
ଵ ቁ ൌ ቀെ^
ଵ ξ
ǡ^
ଶ ξ^
ǡ െ
ଵ ξ
(b)
First use Gram
−Schmidt to get an orthogonal basis:
^ ൌ ܝ
ଵ
^ ൌ ܝ
ۃ^ ܝ
ܞ ǡ
ۄ^
ԡ^ ܞ
ʹ^
ܞ^ ͳ
ଶ^ Ͳ
ଶ^ ^
Chapter 6: Inner Product Spaces
ۃ^ ܝ
ܞ ǡ
ԡ^ ܞ
ԡͳ^
ʹ^
ܞ^ ͳ
ۃ^ ܝ
ܞ ǡ
ԡ^ ܞ
ԡʹ^
ʹ^
ܞ^ ʹ
ଶ^ Ͳ
ଶ^
ଶ^ Ͳ
ଶ^ ʹ
ଶ^ ^
Then an orthonormal basis is
ܙ^
ܞଵ ԡ ܞ^ ͳ
ܙ^
ܞଶ ԡ ܞ^ ʹ
ܙ^
ܞଷ ԡ ܞ^ ͵
Let
ͳ
ൌሺͳǡͲǡ
ͳሻ
. Then
ͳǡͳǡ͵ሻʹሺͳǡͲǡ
ͳሻൌሺͳǡͳǡͳሻ
, and
ൌሺͲǡͳǡʹሻሺͳǡͲǡ
ͳሻ
ሺͳǡͳǡͳሻൌሺͲǡͲǡͲሻ
. Note that since
ሺͳǡͲǡ
ͳሻሺ
ͳǡͳǡ͵ሻൌሺͲǡͳǡʹሻ
, the subspace is
generated by just
ሺͳǡͲǡ
ͳሻ
and
ͳǡͳǡ͵ሻ
, so an orthonormal basis is given by
ǡͳ
Denoting by
the plane spanned by the vectors
and
and noting that these vectors areଶ
orthonormal we obtain ܟ
ௐ^
ܝ ǡܟۃ ൌ ܟ
ܝ ǡܟۃ ଵ
Ͷ Ͳ െ ͷ
ͻ൰ ൬ͷ
Ͷǡ Ͳǡ െ ͷ
͵൰ ሺͲ ʹ ͲሻሺͲǡ ͳǡ Ͳሻͷ
ସ ǡ ʹǡହ^
ଷ ቁହ^
ൌ ሺͳǡ ʹǡ ͵ሻ െ ቀെଵ
ସ ǡ ʹǡହ^
ଷ ቁ ൌ ቀହ^
ଽ ǡ Ͳǡହ^
ଵଶ ହ^
First, we find an orthonormal basis for the space spanned by
ͳ
and
. Letʹ
1
(^1)
^ ^
ͳͳ ͳ^
ʹ^
ʹ
ൌ^
ൌ^
ǡ Ͳǡ
and
2
2 3
3
^
^
^
ʹ^
ʹ^
ͳ^
ͳ
ǡ^
^
ൌ^
ǡ ͳǡ
, so
11
2
2
11 22
22
2
3
3
3
2
3
ǡ ͳǡ
ൌ^
ǡ^
ǡ^
Then, the orthogonal vector projection of
on the space spanned by
ͳ
and
or
ͳ
and
is
11
11
2
2
11
2
2
2
2
1
1
2
2
3
3
1
^
ͳ^
ͳ^
ͳ^
ʹ^
ʹ
ǡ^
^
ǡ^
^
ൌ^
ǡ Ͳǡ
ǡ ͳǡ
ൌ^
ሺͳ͵ǡ ͳǡ ͵ͷሻǤ
^
^
Then
ͳൌ
(^1 )
ʹǡǡ
. Note
ʹ^ is orthogonal to
ͳ
and
ʹ, and
(^1 ) ሺͳ͵ǡͳǡ͵ͷሻ
(^1 )
ʹǡǡ
ʹሻൌሺͳǡʹǡ͵ሻ
Chapter 6: Inner Product Spaces
In order to apply Theorem 6.3.4(a) we need an orthogonal basis for
W
, which
ܝ ǡ
is not. So
we apply Gram-Schmidt:^ ܞ
ൌ ሺെͳǡ Ͳǡ ͳǡ ʹሻଵ
ܞଶ^
ܝ ൌ
െ ଶ^
ܝۃ^ ଶ
ܞ ǡ ଵ
ۄ ܞԡ
ଶԡ (^) ଵ
ܞ
ൌ ሺͲǡ ͳǡ Ͳǡ ͳሻ െଵ
Ͳ Ͳ Ͳ ʹ ሺെ
ͳ^ ሻ
ʹ^
^
ʹͳ ^
ሺെͳǡ Ͳǡ ͳǡ ʹሻ ൌ ሺͲǡ ͳǡ Ͳǡ ͳሻ െ
ͳሺെͳǡ Ͳǡ ͳǡ ʹሻ͵
ଵǡ ͳǡ െଷ^
ଵ ǡଷ^
ଵ ଷ
Now,
is the projection ofଵ
w
on
W
, which by Theorem 6.3.4(a) is
^ ܟ
ۃ^ ܞ ǡܟ
ԡ^ ܞ
ԡͳ^
ʹ^
^ ͳ
ۃ^ ܞ ǡܟ
ԡ^ ܞ
ԡͳ^
ʹ^
^ ʹ
ൌ^
െ͵ Ͳ ʹ ͳͲ ሺെͳሻ
ଶ^ Ͳ
ଶ^ ͳ
ଶ^ ʹ
ሺെͳǡ Ͳǡ ͳǡ ʹሻଶ (^)
ͳ െ ͶȂ
ଶ ାଷ
ఱ య
ሺͳ ͵
ଶ^ ͳ
ଶ^ ሺെ ͳ ͵
ଶ^ ሺͳ ͵
ͳǡ ͳ െ ͵
ͳǡ Ǥ ͵
ͳ൰͵
͵ሺെͳǡ Ͳǡ ͳǡ ʹሻ െʹ
ͳǡ ͳ െ ͵
ͳǡ Ǥ ͵
ͳ൰͵
͵ሺെͳǡ Ͳǡ ͳǡ ʹሻ െ ሺͳǡ ͳǡ െͳǡ ͳሻ ൌ ൬െ ʹ
ͷǡ െͳǡ Ǥ ʹ
ͷǡ ʹ൰ʹ
Thus
ͷǡ െͳǡ ʹ
ͷǡ ʹ൰ ǡ ܟʹ
ൌ ሺ͵ǡ െͶǡ ʹǡ ͷሻ െ ൬െ ଵ
ͷǡ െͳǡ ʹ
ͷǡ ʹ൰ ൌ ൬ʹ
ͳͳʹ
ǡ െ͵ǡ െ
ͳǡ ͵൰ʹ
92. (a)
The columns of the matrix
§^
¨^
^
ʹ^
ͳ
ൌ^
ͳ^
ʹ^
are already orthogonal, so just divide by the norm
ͷ^
to get
. For
, note
5
5 2
(^1)
^
ሺʹǡ
ͳሻ
ǡ^
ͷ^ ,
5
5 2
(^1)
ሺͳǡ ʹሻ
ǡ^
ൌ Ͳǡ
and
5
5 1
2
ሺͳǡ ʹሻ
ǡ^
ൌ^
ͷ^. Thus,
5
5 5
5 2
1 1
2
§^
¨^
¨^
¨^
^
©^
©^
ͷ^
ൌ^
Ͳ^
ͷ^
(b)
Let the normalized vector corresponding to the first column be
ͳ
ൌሺͲǡͳǡͲሻ
. Then
ͳ
ʹ^
ሺͳǡͲǡͳሻ
. Also,
ሺͲǡʹǡͲሻ
ሺͲǡͳǡͲሻൌʹǡሺͳǡͳǡͳሻ
ሺͲǡͳǡͲሻൌͳ
, and
ሺͳǡͳǡͳሻ
¨^
©^
ͳ^
ͳ ǡ Ͳǡʹ
ʹ^
ൌ^
ʹ^. So,
§^
¨^
¨^
¨^
¨^
¨^
¨^
¨^
©^
ͳ Ͳ^
ʹ^
ʹ^
ͳ
ͳ
Ͳ^
Ͳ^
ͳ Ͳ^
Best Approximation; Least Squares 99.
(a)
The normal system
is ͷ^ ቂ
Ͳ^
ቃቂͻ
Ͷെͻ
Chapter 6: Inner Product Spaces
The reduced row echelon form of the augmented matrix of this system is
ͳ ቈ
Ͳ^
Ͳ^
ͳ^
െͳ
Ǥ^
The
solution of this system,
ସ ହ^ ݔ ǡ
ଶ^
ൌ െͳ
is the unique least squares solution of
A x
=^
b.
(b)
The normal system
is ͻ ቂ
െͳ െͳ
The reduced row echelon form of the augmented matrix of this system is
ͳ
Ͳ^
ଵହଷ
Ͳ^
ͳ^
െ^
ଵହହଷ
Ǥ^
The
solution of this system,
ଵ ହଷ
ݔ ǡ
ଵହ ହଷ
is the unique least squares solution of
A
x^ =
b.
100. (a)
The normal system is
§^
·^
§^
§^
·^
§^
¨^
¸^
¨^
¨^
¸^
¨^
¨^
¸^
¨^
©^
¹^
©^
¨^
¸^
¨^
©^
¹^
©^
ͳ^
ʹ^
ͳ^
ͳ^
ʹ^
ͳ^
ͳ^
ͳ^
ͳ^
ൌ^
ʹ^
ͳ^
Ͳ^
ʹ^
ͳ^
ʹ^
Ͳ^
ͳ
^
, or
§^
·^
§^
¨^
¸^
¨^
©^
¹^
©^
^
͵^
͵^
ͷ^
^
, so
§^
§^
¨^
¨^
©^
ͷ^
͵^
^
ͳͳ
ͳ^
ͳ
ൌ^
͵^
^
ͺ^
ʹͳ
ʹͳ
^
(b)
The normal system is
§^
·^
§^
^
^
^
§^
·^
§^
¨^
¸^
¨^
¨^
¸^
¨^
¨^
¸^
¨^
^
¨^
¸^
¨^
¨^
¸^
¨^
¨^
¸^
¨^
¨^
¸^
¨^
©^
¹^
©^
^
©^
¹^
©^
ͳ^
ͳ^
Ͳ^
ͳ^
ʹ^
ͳ^
ͳ^
ͳ^
ʹ^
ͳ^
ͳ
ʹ^
ͳ^
ͳ^
ͳ
ͳ^
ͳ^
Ͳ^
ͳ^
ൌ^
ͳ^
ͳ^
Ͳ^
ͳ
ͳ^
Ͳ^
ͳ^
Ͳ^
ͳ^
ͳ^
ͳ^
Ͳ^
ͳ^
ͳ^
ͳ
ͳ^
ͳ^
ͳ^
ͳ
^
which is
§^
·^
§^
¨^
¸^
¨^
¨^
¸^
¨^
¨^
¸^
¨^
¹^
©^
^
Ͳ^
ʹ^
ͳ
Ͳ^
͵^
Ͳ^
ൌ^
ʹ^
Ͳ^
^
. So,
13 §^
·^
§^
§^
¨^
¸^
¨^
¨^
¨^
¸^
¨^
¨^
¨^
¸^
¨^
¨^
©^
©^
͵^
ʹ^
ͳͳ
ͳ^
ͳ^
ͳʹ͵͵Ͳ
ʹ^
ͳ^
ͳ^
ͳ
Ͳ^
ͳ
ൌ^
Ͳ^
Ͳ^
ʹ^
Ͳ^
^
101. (a)
͵^
ͳ^
ͳ
ʹ^
Ͷ^
ͳͲ
െͳ
͵^
͵^
ʹ^
െͳ
ͳ^
Ͷ^
ͳ^
ͳͲ
ͳͳ
ͳʹͲ
ͷͻ
͵^
ͳ^
ͳ
ʹ^
Ͷ^
ͳͲ
െͳ
͵^
ͳെʹ൩ ൌ ͳ
െͳͶ
The normal system is
்ܣ^
or
ͳͳ
ͳʹͲ
ͷͻ
െͳͶ
The reduced row echelon form of the augmented matrix of the normal system is
ͳ ۍ ێ ێ ێ ۏ
Ͳ^
Ͳ^
^
ଶ
Ͳ^
ͳ^
Ͳ^
Ͳ^
Ͳ^
ͳ^
െ^
Chapter 6: Inner Product Spacessolution of
is
െ^
Denoting
ܞ ǡଵ
ሽ^ ଶ
we obtain
ௐ^
െͳ
ʹ^
ͳ^
െ^
͵൩െͶെͳ
106. (a)
Let the
be the columns of a matrix
^
and solve find the least squares solution:
¨^
¨^
¨^
©^
ͷ͵
ͳ
ͷ͵
ൌ^
ൌ^
ͳ
^
^
. The projection is then
§^
§^
·^
¨^
¨^
¸^
¨^
¸^
¨^
¨^
¸^
¨^
¨^
¸^
¨^
¨^
©^
¹^
©^
ͷ͵ ¹
ͷ͵^
ʹ͵ͺ͵
ͷ͵
ͳ^
Ͳ^
ʹ^
ͳ^
ͳ
ൌ^
ͳ^
ͳ^
ͳ^
ͳ^
Ͳ^
ͳ^
107. (a)
§^
¨^
¸^
z
¨^
¨^
©^
ͳͶ
ͷ^
ͷ
ʹ^
ͳ^
ͳ^
, so
is invertible, and hence the columns of
^
are
linearly independent.
108. (a)
Letting
ͳቃͲ
, we have
ܲሾ^
ଵ^
்ܣ^
ͳቃ ቀሾͳͲ
ͳቃቁିͲ
ଵ
ሾͳ
ͳቃ ሺሾͳሿሻିͲ
ଵ^
ሾͳ
ͳቃ ሾͳሿሾͳͲ
ͳቃ ሾͳͲ
ͳ^
Ͳ^
(b)
Letting
Ͳቃͳ
, we have
ܲሾ^
ଵ^
்ܣ^
Ͳቃ ቀሾͲͳ
ͳሿ ቂ
Ͳቃቁିͳ
ଵ
ͳሿ ൌ ቂ
Ͳቃ ሺሾͳሿሻିͳ
ଵ^
ͳሿ ൌ ቂ
Ͳቃ ሾͳሿሾͲͳ
ͳሿ
Ͳቃ ሾͲͳ
ͳሿ ൌ ቂ
Ͳ^
Ͳ^
ቃͳ
109. (a)
A basis for the
xy
-plane is
ǡͳ
^ ʹ
ሽ, so we use these as the columns to get
§^
¨^
¨^
¨^
©^
ͳ^
ൌ^
Ͳ^
ͳ Ͳ^
Then,
and
§^
·^
§^
§^
¨^
¸^
¨^
¨^
¨^
¸^
¨^
©^
©^
¹^
©^
ͳ
ͳ^
Ͳ^
ͳ^
Ͳ^
ͳ^
Ͳ^
ൌ^
ͳ^
ͳ^
Ͳ^
ͳ^
Ͳ^
Ͳ^
Ͳ^
Ͳ^
^
Chapter 6: Inner Product Spaces
Letting
, Formula (10) in Section 6.4 yields
ܲሾ^
ଵ^
ଵ
ܾܿܽሾ^
ଶ
ଶ^ ܿ
ଶ
ଵሿି^
ൌ^
ଵ మ ା
మ^ ା
ܾܾܾܽܿܽܽଶ
ܾܾܿܿܿܽܿଶ
111. (a)
A basis for the plane
can be found by taking 3 points on the plane
ሺͶǡ͵ǡͲሻ
ͳǡͲǡ͵ሻ
, and
ሺͲǡͳǡͶሻ
, and finding the vectors between them: basis is
·^
§^
°^
¨^
¸^
¨^
®^
¨^
¸^
¨^
°^
^
©^
¹^
©^
¯^
ͷ^
͵ ǡ
͵^
(b)
Make these basis vectors the columns of
^
and we get
17
6
3
6
5
2
3
2
25
§^
¨^
¨^
¨^
¨^
ͳ
ʹ^
ͳ͵^
ʹ
ͳ͵^
ͳ͵^
ͳ͵
ʹ^
ͳ͵^
ʹ
ൌ^
^
(c)
§^
¨^
¨^
¨^
¨^
^
©^
^
^
^
^
^
^
ͳ^
^
͵
ʹ^
ͳ͵^
ʹ
^
ͷ^
ʹ
ͳ͵^
ͳ͵^
ͳ͵
͵^
ʹ^
ʹͷ
ʹ^
ͳ͵^
ʹ
ൌ^
(d)
So, if
ൌሺʹǡͳǡ
ͳሻ
49 15 27
§^
¨^
¨^
¨^
¨^
ʹ ͳ͵ ʹ ൌ
Thus the distance between the point
ሺʹǡͳǡ
ͳሻ
and its
projection is
͵^
ʹ^
ͳ^
ͳ
ʹ^
ʹ^
ʹ^
ʹ
ǡ^
ǡ^
(which agrees with the formula for finding the distance
between a point and a plane).
112. (a)
ሺʹǡ െͳǡ Ͷሻሽ
so that the vector
ሺʹǡ െͳǡ Ͷሻ
forms a basis for
(its linear
independence follows from Theorem 4.3.2(b))
Chapter 6: Inner Product Spaces
(b)
Letting
ʹെͳ൩ Ͷ
, Formula (10)
of Section 6.4 yields
ܲሾ^
ଵ^
்ܣ^
െͳ
െͳ
ʹ൩൱ିെͳͶ
ଵ
െͳ
ʹെͳ൩ ሾʹͳሿିͶ
ଵ^
ଵ ଶଵ
Ͷ^
ͳ^
ͺ^
ͳ
(c)
ଵଶଵ^
Ͷ^
ͳ^
ͺ^
ͳ
ۍ^ ێێێۏ
ସݔଶଵ^
ଶ ଶଵ
ଶଵ
െ^
ଶݔଶଵ^
ଵ ଶଵ
ସ ଶଵ
ଶଵ
ସ ଶଵ
ଵ ଶଵ
(d)
ଵ ଶଵ
Ͷ^
ͳ^
ͺ^
ͳ
ʹͳ൩ ൌെ͵
ଷ െ^
ଵଶ
ېۑ^ ; the distance betweenۑۑے
and
equals to the distance
between
and its projection on
݀൬ ሺʹǡ ͳǡ െ͵ሻǡ ቀെ
ǡ^
ଷ ǡ െ
ଵଶ ^
ቁ^
ଶ ቀͳ െ
ଷ ቁ^
ଵଶ
ଵ^
By inspection we see that if
ͳ,ʹ
the point on line
l^ is
^
ͳ^
ͳ ʹ^
ʹ
ͳǡ^
ǡ^
and if
ͳʹ
the point on
line
m
is
^
ͳ^
ͳ ʹ^
ʹ ͳǡ^
ǡ^
. Since the lines intersect, we've found the values of
^ and
^ that minimize
the distance because for these two points, the distance is
Least Squares Fitting to Data 119.
If the three points were on the line
y^
=^ a
+^
bx
, they would give the system:
ͳ
a^
b a^
b a^
b
Chapter 6: Inner Product Spaces
Letting
A
§^
¨^
¨^
¨^
©^
ͳ^
ൌ ͳ
ͳ^
we get
11 15
§^
·^
§^
§^
·^
¨^
^
¨^
¨^
¸^
¨^
¸^
¨^
©^
¹^
¨^
¸^
©^
©^
ͳ^
ͳͶ ͳͶ
ͳ^
^
a b^
, or
11
15
^ ͳͶ
ͳͶ
We have
ͳ^
ͳ^
ͳ^
ͳ^
,^ ்ܯ
Ͷ^
ͺ^
ቃ,^
ଵ^
ൌ^
ଵቂ ଶସ^
, and
ଵ^
ଵ ଶସ
ͳ^
ͳ^
ͳ^
ͳ
Ͳ^
ʹ^
͵^
ͳͲ൪ ൌ ͳʹ
so the least squares straight line
fit to the given data points is
ଶ ଷ^
^
ଵݔ^
The system we want to approximate is
§^
·^
§^
§^
¨^
¸^
¨^
¨^
¨^
¸^
¨^
¨^
¨^
¸^
¨^
^
¨^
¨^
¸^
¨^
©^
©^
¹^
©^
ͳ^
ͳ^
ͳ^
ͳ^
ʹ^
Ͷ^
ͳ ൌ
ͳ^
ͳ^
ͳ^
ͷ
ͳ^
ʹ^
Ͷ^
a b c
. Our approximation is
^
§^
·^
§^
§^
·^
¨^
¸^
¨^
¨^
¸^
¨^
¸^
¨^
¨^
¸^
¨^
¨^
¸^
¨^
¨^
©^
¹^
©^
©^
Ͷͳ
ͳ^
ͳͳͲͶ͵
ͳ
ൌ^
ൌ^
ͷ ʹ
a b c
So the quadratic polynomial is
^
^
Ͷͳ^
ͳ^
Ͷ
^
ͳͲ^
͵
We have
ͳۍ ێێێۏ
െͳ
ሺെͳሻ
ଶ^
ሺെͳሻ
ଷ
ͳ^
Ͳ^
ͳ^
ͳ^
ଶͳ
ଷͳ
ͳ^
ʹ^
ͳ^
͵^
ͳۍێێێۏ
െͳ
ͳ^
െͳ
ͳ^
Ͳ^
Ͳ^
ͳ^
ͳ^
ͳ^
ͳ
ͳ^
ʹ^
Ͷ^
ͳ^
͵^
ͻ^
ېۑ^ ,ۑۑ
ͷ^
ͷ^
ͳͷ
͵ͷ
ͷ^
ͳͷ
͵ͷ
ͻͻ
ͳͷ
͵ͷ
ͻͻ
ʹͷ
͵ͷ
ͻͻ
ʹͷ
ͻͷ
൪,^
ଵ^
ۍ^ ێێێێێێۏ
ଶଷହ
െ^
ଵ ସଶ^
െ^
ସ
ଵ
െ^
ଵ ସଶ
ଶହଷ
െ^
ହ଼ ସ^
െ^
ଵ ଷ
െ^
ସ ^
െ^
ହ଼ ସ
ଷଽହ
െ^
ହ ଶସ
ଵ ^
െ^
ଵ ଷ^
െ^
ହ ଶସ
, and
ଵ^
ۍ^ ێێێێێێ ۏ
ଶଷହ
െ^
ଵ ସଶ^
െ^
ସ
ଵ
ଵ ସଶ
ଶହଷ
ହ଼ ସ^
െ^
ଵ ଷ
െ^
ସ ^
െ^
ହ଼ ସ
ଷଽହ
െ^
ହ ଶସ
ଵ ^
െ^
ଵ ଷ^
ହ ଶସ
ͳ^
ͳ^
ͳ^
ͳ^
ͳ
െͳ
Ͳ^
ͳ^
ʹ^
ͳ^
Ͳ^
ͳ^
Ͷ^
ͻ
െͳ
Ͳ^
ͳ^
ͺ^
െͳͶെͷെͶ
ېۑۑͳۑ ےʹʹ
െͷ
Thus, the cubic polynomial that best fits the given data points is
ݕൌ െͷ ͵ ݔെ Ͷݔ
ଶ^ ݔʹ
Chapter 6: Inner Product Spaces
ܽ
ଵ గ^
^
ଶ^ ʹݔሻ
݇ ݔ
ଶగ݀^
ଵቀగ^
ଶሺଵା௫ሻ
మ^
ଶయ^ ^
௫ାଶ௫
^
ଶగ
ൌ^
ସమ^
ଵ గ^
^
ଶ^ ʹݔሻ ݇ ݔ
ଶగ݀^
ଵቀെగ^
ሺଵା௫ሻ
మ^ ௦^
ଶయ^ ^
ଶሺ௫ାଵሻ
మ^
ଶగ
ସ మ^
ሺͳ ߨሻ
ss
(a)
ଶ^
ൎ^
^ బଶ^
ଵ
ଷ
ଵ
ଶ
ଷ
so
ߨଷ^
ଶ^ Ͷ
ݔ ʹݔ
ସ
͵ݔଽ^
െͶሺ ߨ ͳሻ ݔ െ ʹሺ ߨ ͳሻ ʹݔ െ
ସ ଷ^ ሺ ߨ ͳሻ ͵ݔ
(b)
ݔ
ଶ^ ൎ
బ ଶ
ܽ
ଵ
ݔ ܽ
ଶ
ʹ ݔ ڮܽ
ܾ ݔ ݊
ݔ ܾଵ
ଶ
ʹ ݔ ڮܾ
ݔ ݊
so
ߨଷ^
ଶ^ Ͷ
ݔ ʹ ݔ ڮ
ସ మ^
െͶሺ ߨ ͳሻ ݔ െ ʹሺ ߨ ͳሻ ʹ ݔെ ڮ െ
ସ ^ ሺ ߨ ͳሻሺݔ݊ሻ
132. (a)
First use Gram-Schmidt to find an orthonormal basis for the space
spanned by
ሼͳǡ
Verify that
®^
¯^
ͳ
ͳǡ
Ƚ^
is such a basis where
^
^
ͳሻ
ൌ^
ͳ ൌ
^
Ƚ^
^
^
Thus, the projection is
^
^
^
^
^
ͳ
ͳ
ሺͳ
ሻ ൌ ൏ͳ
ǡ ͳ ͳ൏ͳ
ǡ^
^
^
§^
·
^
^
^
¨^
¸^
^
©^
¹
^
^
^
^
͵^
͵^
͵^
͵^
͵
ͳ^
ͳ^
ͳ^
ͳ
ʹ^
Ƚ^
ʹ^
ʹ^
ʹ^
ʹ^
ͳ^
ʹ^
ͳ
ͳ
ʹ
ൌ^
^
ൌ^
^
ሺ^
ͳሻ ൌ
ሺ^
ሺ^
ͳሻሻ ൌ
ሺ͵
ሻሺ
ͳሻ
(b)
The mean square error is
^
^
^
ͳ^
ʹ
ͳ^
ͳ ʹ^
ͳ
Ͳ
ͳͻ
ͳ
ൌ ǤͲͲͳ͵ͷ
ͳʹ
ͳʹ
133. (a)
Let us denote
ܹ ൌ ሼͳǡ ݔሽ
. Applying the Gram-Schmidt process to the basis
ൌ ͳଵ
and
we obtain an orthogonal basis
ܞଵ
ൌ ͳ
,^ ܞ
ܝۃ
ܞǡమ
ۄభ ܞԡభ
మ^ ԡ
^ భ௫ௗ௫బ^ ^ భଵௗ௫బ^
ͳ ൌ ݔ െೣ
భమ൨మ బభ௫ሿబ
ଵ ଶ^
and an orthonormal basis
ܞ భܞԡԡభ^
ଵ ට
భଵ (^) బ^
ௗ௫
ൌ^
ଵ భට௫ሿబ
ൌ ͳ
,^ ܙ
ൌଶ^
ܞ మܞԡԡ^ మ^
ൌ^
௫ି^
భమ
ට ቀ௫ି
భ మ మቁ భబ^
ௗ௫
ൌ^
௫ି^
భమ ቀೣషඩ
భ ቁమ య య^
భ^ బ ൌ ʹξ͵ሺݔ െ
ଵ ሻଶ^
The least squares approximation to
௫^ from
is
ௐ^
ܙ ǡۃ ൌ
ܙ ǡۃ ଵ
ܙۄଶ^
௫ ଵ^
ቆන ʹξ͵ ቀݔ െ
ଵ ݁ቁଶ^
௫
ଵ ݀
ݔ^
ቇ ʹξ͵ሺݔ െ
ଵ ሻଶ^
ൌ ݁െ ͳ ൬ʹ
݁ݔቀ ξ͵
௫
ଷ ݁ଶ
௫^ ቁቃ
ଵ൰ ʹξ͵ ቀݔ െ
ଵ ቁ ൌ Ͷ ݁െ ͳͲ ሺ͵ െ ݁ሻݔଶ^
Chapter 6: Inner Product Spaces
(b)
The mean square error is
௫ െ ሺͶ ݁െ ͳͲ ሺ͵ െ݁
ଶ
ଵ݀^
ହ ଶ^
ͲǤͲͲ͵ͻͶ
134. (a)
First use Gram-Schmidt to find an orthonormal basis for the space
spanned by
ሼͳǡ
Verify that
^^
`
͵^
ͷ
ͳ^
ͳ ʹ^
ʹ^
ʹ
ǡʹ
ǡ^
͵^
ͳ
^
is such a basis. Then
³^
ͳ
ͳ^
ͳ
ʹ^
ʹ^
ͳ
ሻǡ^
³^
^
ͳ
͵^
͵
ʹ^
ʹ^
ͳ
ሻǡ^
S
^
^
ʹ
ͳ
ʹ^
ʹ
ͷ^
ͷ
ͳ^
ͳ
ʹ^
ʹ^
ʹ^
ʹ^
Ǧͳ
͵ ͳͲ
ሻǡ^
͵^
ͳ ൌ
͵^
ͳ
Thus
ʹ
ʹ ͵^ ͳͲ
Ɏ
^
(b)
The mean square error is
Ɏ
Ɏ
Ɏ
^
Ͷ
ʹ^
Ͷ
ͳ^
ʹ ͵^ ͳͲ
͵ ʹͶ
ͳͲ
ሺͳ
ൌ ͲǤͷͻ
Using the trigonometric identity
ଵ ଶ^
ଵ ଶ^
we obtain
ǡۃ
ଵ ଶ^
^
ଶగ݀^
ଵଶ^
ଶగ݀^
ݔ^
where both
and
are
nonzero integers.Substituting
in the first integral, and
in the second integral yields
ǡۃ
ଵ ଶ ୱ୧୬ሺሺି
ሻ௫ሻ ି
െ^
ଵୱ୧୬ሺሺାሻ௫ሻଶ
ା
since
for any integer
Use Formulas (8) of Section 6.6, and integration by parts to integrate
cos
݇ ݔ^
and
sin k
ǣݔ
ଵ గ
^
ଶగ݀^
ଵ ݔߨቀగ^
ଵ ଶ^
ଶగ
ଵ గ^
^
ଶగ݀^
ଵቀగ^
ଵ మ^
గା௫
ଶగ
ଵ గ
^
ଶగ݀^
ଵቀെగ^
గା௫ గ
ଵ݇ ݔቁ ቚమ^ ^
ଶగ
ଶ
The Fourier series for
over the interval
ሾͲǡ ʹߨሿ
is
ଶݔ݇^
ஶ ୀଵ
Let
d൏
S^ and
S
d
d
ʹ^ S
. Then
Chapter 6: Inner Product Spaces
k ª^
^
¬^
³^
³^
³^
a a
ʹ^
ʹ
Ͳ^
Ͳ ʹ^
ʹ
Ͳ ʹ^
ʹ
Ͳ ͳ^
ͳ
ሺ^
ሻ^
ൌ^
ൌ ͳǡ
ͳ^
ͳ
ሺ^
ሻ^
ൌ^
ሻ^
ൌ ͳǡ ʹǡ ǤǤǤ
ͳ^
ͳ^
ͳ
ሺ^
ሻ^
ൌ^
ሻ^
ൌ^
ሺ ͳሻ
ͳ Ǥ
Ɏ
Ɏ Ɏ
Ɏ
Ɏ
Ɏ
Ɏ
Ɏ
Ɏ
^
^
^
^
^
So, the Fourier series is
f^
f^
ª^
^
¬^
¦
¦
ͳ^
ͳ^
ʹ
ʹ^
ʹ^
ሺʹ^
ͳሻ
ൌͳ^
ൌͳ
ͳ
ሺ ͳሻ
ͳ ሺ
ሻ^
ͳሻ
^
Ɏ
^
The Fourier series of cos(
) is just cos(
). That is,
a^3
= 1, and all other coefficients are zero.
