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w s o - - PowerPoint PPT Presentation


slide-1
SLIDE 1
  • w
slide-2
SLIDE 2
  • s
slide-3
SLIDE 3

⃗ , . . . ,⃗ ∈ R V R ⃗ , . . . ,⃗ V ∈ R× ∥⃗ − ⃗ ∥

= ∥⃗

−⃗ ∥

.

˜ = ⃗ V R

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slide-4
SLIDE 4

⃗ , . . . ,⃗ V ∈ R× = =

, . . . ,⃗ V

⃗ , . . . ,⃗ ∈ R ∈ R× ⃗ , . . . ,⃗ ∈ R V ∈ R× ⃗ , . . . ,⃗

  • c

⇐ × ,

fifth swig,

slide-5
SLIDE 5

⃗ , . . . ,⃗ V ∈ R× = = () ≤ ⃗ , . . . ,⃗ V

⃗ , . . . ,⃗ ∈ R ∈ R× ⃗ , . . . ,⃗ ∈ R V ∈ R× ⃗ , . . . ,⃗

¥÷÷k÷÷ ÷÷÷:*"

CI-wily

slide-6
SLIDE 6

⃗ , . . . ,⃗ V ∈ R× = = () ≤ ⃗ , . . . ,⃗ V

⃗ , . . . ,⃗ ∈ R ∈ R× ⃗ , . . . ,⃗ ∈ R V ∈ R× ⃗ , . . . ,⃗

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n x a

n ,

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wig i s closed-I'

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point

t o y ' n ' t

a s i t s i t ' m

subspace

I = x v

X a x w t

X v

slide-7
SLIDE 7

()()⃗ = ()⃗ ⃗ ∈ R ( − ) = ⃗ ∈ R ∥⃗ ∥

= ∥()⃗

+ ∥⃗

− ()⃗ ∥

.

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projection 1 you

a r e

i n subspace

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projectpointsi n

subspace,

t w

nothing

should happen

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v t v = I

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w i g

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a

thirty-will

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high Huviylli

⇐ ± H

slide-8
SLIDE 8

⃗ , . . . ,⃗ V R

  • w t idcI

H v } x

Hatbllis Hallet

1161122

t h a t

Hatbllithallium

⇐ '1*1=4,11

i n

general

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Proof o f Pythagorean

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Theoren: hyper:Ilwtylli'

t

11ft-vnight

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llyllisytyllysllwty.to-wily,I l i a

m i

= (wjytlI.ws/ayoJT(vvIytCI-

wjTyg]

I

① yiwtvvtytytce.ru'TLI-wilyt

2yt4yTlI-wily

m y t h

+ "¥wYylli

slide-9
SLIDE 9

⃗ , . . . ,⃗ V R

  • T x
slide-10
SLIDE 10

⃗ , . . . ,⃗ V R ⃗ , . . . ,⃗ V ∈ R× ⃗ ∈ R ∈ R

slide-11
SLIDE 11

⃗ , . . . ,⃗ V R ⃗ , . . . ,⃗ V ∈ R× ⃗ ∈ R ∈ R

slide-12
SLIDE 12

⃗ , . . . ,⃗ V R ⃗ , . . . ,⃗ V ∈ R× ⃗ ∈ R ∈ R V

slide-13
SLIDE 13

⃗ , . . . ,⃗ ∈ R

slide-14
SLIDE 14

⃗ , . . . ,⃗ ∈ R

slide-15
SLIDE 15

⃗ , . . . ,⃗ ∈ R

  • Hi-xilli=

L

A

( E )

a

2 8

¥÷÷'' }

'¥"

  • -I
slide-16
SLIDE 16

⃗ , . . . ,⃗ ∈ R

slide-17
SLIDE 17

⃗ , . . . ,⃗ ∈ R

  • x

x

x x x

i f

X

i s

a nK

K

slide-18
SLIDE 18

⃗ , . . . ,⃗ ∈ R

  • v
slide-19
SLIDE 19

⃗ , . . . ,⃗ ∈ R

slide-20
SLIDE 20

⃗ , . . . ,⃗ ∈ R

slide-21
SLIDE 21

⃗ , . . . ,⃗ ∈ R

  • X IXu's

E

  • ,

I

slide-22
SLIDE 22

⃗ , . . . ,⃗ V ∈ R× V

, . . . ,⃗ ∈ R ∈ R× ⃗ , . . . ,⃗ ∈ R V ∈ R× ⃗ , . . . ,⃗

  • f
  • n'xd lmk
slide-23
SLIDE 23

⃗ , . . . ,⃗ V ∈ R× V V

, . . . ,⃗ ∈ R ∈ R× ⃗ , . . . ,⃗ ∈ R V ∈ R× ⃗ , . . . ,⃗

slide-24
SLIDE 24

⃗ , . . . ,⃗ V ∈ R× V V

∈R× ∥ − ∥ =

  • ,

(, − (),) =

  • =

∥⃗ − ⃗ ∥

, . . . ,⃗ ∈ R ∈ R× ⃗ , . . . ,⃗ ∈ R V ∈ R× ⃗ , . . . ,⃗

  • Eigo

Exinging

  • '÷-÷⇐÷*.
slide-25
SLIDE 25

⃗ , . . . ,⃗ V ∈ R× V V

∈R× ∥∥ − ∥∥ =

  • =

∥⃗ ∥

− ∥⃗

, . . . ,⃗ ∈ R ∈ R× ⃗ , . . . ,⃗ ∈ R V ∈ R× ⃗ , . . . ,⃗

  • riginal projected

Hxi-Wtxilli

d a t -

n

slide-26
SLIDE 26

⃗ , . . . ,⃗ V ∈ R× V V

∈R× ∥∥ − ∥∥ =

  • =

∥⃗ ∥

− ∥⃗

, . . . ,⃗ ∈ R ∈ R× ⃗ , . . . ,⃗ ∈ R V ∈ R× ⃗ , . . . ,⃗

slide-27
SLIDE 27

⃗ , . . . ,⃗ V ∈ R× V V

∈R× ∥∥ =

  • =

∥⃗ ∥

, . . . ,⃗ ∈ R ∈ R× ⃗ , . . . ,⃗ ∈ R V ∈ R× ⃗ , . . . ,⃗

  • if
slide-28
SLIDE 28

∥ − ∥

∈R× ∥∥ =

  • =

∥⃗ ∥

, . . . ,⃗ ∈ R ∈ R× ⃗ , . . . ,⃗ ∈ R V ∈ R× ⃗ , . . . ,⃗

slide-29
SLIDE 29

∥ − ∥

∈R× ∥∥ =

  • =

∥⃗ ∥

, . . . ,⃗ ∈ R ∈ R× ⃗ , . . . ,⃗ ∈ R V ∈ R× ⃗ , . . . ,⃗

  • inarspspj.fi#llvtwTxills

1lvvtxil11NIIz2=llvtxilli

  • llvvtxillisxitwivvtx

isxitvvtxi.tl/vtxilli

slide-30
SLIDE 30

∥ − ∥

∈R× ∥∥ =

  • =

∥⃗ ∥

=

  • =
  • =

⟨⃗ ,⃗ ⟩

⃗ , . . . ,⃗ ∈ R ∈ R× ⃗ , . . . ,⃗ ∈ R V ∈ R× ⃗ , . . . ,⃗

  • cut:#¥i÷÷

¥±.''

slide-31
SLIDE 31

∥ − ∥

∈R× ∥∥ =

  • =

∥⃗ ∥

=

  • =
  • =

⟨⃗ ,⃗ ⟩

⃗ , . . . ,⃗ ∈ R ∈ R× ⃗ , . . . ,⃗ ∈ R V ∈ R× ⃗ , . . . ,⃗

slide-32
SLIDE 32

∥ − ∥

∈R× ∥∥ =

  • =

∥⃗ ∥

=

  • =
  • =

⟨⃗ ,⃗ ⟩

⃗ , . . . ,⃗ ∈ R ∈ R× ⃗ , . . . ,⃗ ∈ R V ∈ R× ⃗ , . . . ,⃗

  • .
  • ¥
slide-33
SLIDE 33

∥ − ∥

∈R× ∥∥ =

  • =

∥⃗ ∥

=

  • =
  • =

⟨⃗ ,⃗ ⟩

⃗ , . . . ,⃗ ∈ R ∈ R× ⃗ , . . . ,⃗ ∈ R V ∈ R× ⃗ , . . . ,⃗

slide-34
SLIDE 34
slide-35
SLIDE 35
  • ∥∥
  • ∈ R×