mat lab tutorial and linear algebra review

7/23/2019 Mat Lab Tutorial and Linear Algebra Review

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Hyoungjune Yi: [email protected]

• Homework at the end of class.

• Text is on resere in the C! li"rary.

• #ower$oint should "e aaila"le "y%&am class day.

mailto:[email protected]






'atla" tutorial and (inear

Alge"ra )eiew• Today*s goals:• (earn enough matla" to get started.

• )eiew some "asics of (inear Alge"ra• +ssential for geometry of $oints and lines.• ,ut also- all math is linear alge"ra.• ok slight exaggeration/.

• 'any slides today ada$ted from 0ctaiaCam$s- #enn !tate.



!tarting 'atla"

• 1or #Cs- 'atla" should "e a $rogram.

• 1or !un*s:

2umerical

Analysis and 3isuali4ation

'atla" 5.%



Hel$

• hel$

• hel$ command

+g.- hel$ $lus

• Hel$ on tool"ar

• demo

• Tutorial:htt$:66amath.colorado.edu6scico6tutorials6matla"6



'atla" inter$reter

• 'any common functions: see hel$ o$s



3ectors

• 0rdered set of

num"ers: %-7-8-9/

• +xam$le: x,y,z /coordinates of $t in

s$ace. r unit vectoais,1If

),(

1

2

,,21

vv

xv

x x xv

n

i i

n

==

=

∑=



ndexing into ectors



3ector Addition

),(),(),(22112121

y x y x y y x x ++=+=+wv

vvww

V+wV+w



!calar #roduct

),(),(2121

axax x xaa ==v

vv

avav



0$erations on ectors

• sum

• max- min- mean- sort- ;

• #ointwise: .<



nner dot/ #roduct

vv

ww

αα

22112121 .),).(,(. y x y x y y x xwv +==

The inner $roduct is aThe inner $roduct is a !CA(A)=!CA(A)=

α cos||||||||),).(,(. 2121 wv y y x xwv ⋅==

wvwv ⊥⇔= 0.



'atrices

=×

nmnn

m

m

m

mn

aaa

aaa

aaa

aaa

A

21

33231

22221

11211

mnmnmn B AC ××× +=Sum:Sum:

ijijij bac +=

A and , must hae the same A and , must hae the same

dimensionsdimensions



'atrices

pmmn pn B AC ××× =Product:Product:

∑==

m

k

kjik ij bac1

A and , must hae A and , must hae

com$ati"le dimensionscom$ati"le dimensions

nnnnnnnn A B B A ×××× ≠

dentity 'atrix:

A AI IA I ==

=

100

010

001



'atrices

mnT

nm AC ×× =Trans$ose:Trans$ose:

jiij ac = T T T A B AB =)(

T T T B A B A +=+ )(

f f A AT = A is symmetric A is symmetric



'atrices

Determinant:Determinant: A must be squareA must be square

3231

2221

13

3331

2321

12

3332

2322

11

333231

232221

131211

detaa

aaa

aa

aaa

aa

aaa

aaa

aaa

aaa

+−=

12212211

2221

1211

2221

1211det aaaa

aa

aa

aa

aa−==



'atrices

I A A A A nnnnnnnn == ××−

×−

×11

nerse:nerse: A must "e s>uare A must "e s>uare

−

−−

=

−

1121

1222

12212211

1

2221

1211 1

aa

aa

aaaaaa

aa



ndexing into matrices



+uclidean transformations



7? Translation

tt

PP

P’P’



7? Translation +>uation

PP

xx

y y

ttxx

tt y y

P’P’tt

tPP +=++= ),(' y x t yt x

),(

),(

y x t t

y x

=

=

t

P



7? Translation using 'atrices

PP

xx

y y

ttxx

tt y y

P’P’tt

),(

),(

y x

t t

y x

=

=

t

P

⋅

=

+

+

→1

1

0

0

1

' y

x

t

t

t y

t x

y

x

y

x

P

tt PP



!caling

PP

P’P’



!caling +>uation

PP

xx

y y

s.xs.x

P’P’s.ys.y

),('

),(

sy sx

y x

=

=

P

P

PP ⋅= s'

⋅

=

→ y

x

s

s

sy

sx

0

0

'P

SPSP ⋅='



)otation

PP

P’P’



)otation +>uations

Counter-clocwise rotation by an an!leCounter-clocwise rotation by an an!le θθ

−

=

y

x

y

x

θ θ

θ θ

cossin

sincos

'

'

PP

xx

"’"’P’P’

θθ

#’#’

y y R.PP'=



?egrees of 1reedom

$ is %x%$ is %x% & elements& elements

'()* )ere is only , de!ree o reedom:'()* )ere is only , de!ree o reedom: θθ

1)det( ==⋅=⋅

R

IR R R R TT

)e & elements must satisy te ollowin! constraints:)e & elements must satisy te ollowin! constraints:

−=

y

x

y

x

θ θ

θ θ

cossin

sincos

'

'



!tretching +>uation

PP

xx

y y

SSxx.x.x

P’P’SS y y.y.y

⋅

=

→ y

x

s

s

y s

x s

y

x

y

x

0

0'P

),('),( y s x s

y x

y x==

P

P

S

PSP ⋅='



!tretching tilting and $rojecting

with weak $ers$ectie/

⋅

=

⋅

=

→

y

x s

s

s

y

x

s

s

y s

x s

y

x

y y

x

y

x

10

0

0

0'P



(inear Transformation

−

⋅

−

=

⋅

−

−

=

⋅

→

y

x s

s

s

y

x

s

s

y

x

d c

ba

y

x

y

y

x

ϕ ϕ

ϕ ϕ

θ θ

θ θ

ϕ ϕ

ϕ ϕ

θ θ

θ θ

sincos

cossin

10

0

sincos

cossin

sincos

cossin

0

0

sincos

cossin

'P!3?



Affine Transformation

⋅

→1

' y

x

tyd c

txba

P



1iles



1unctions

• 1ormat: function o testx-y/

• 2ame function and file the same.

• 0nly first function in file is isi"le

outside the file.



mages



?e"ugging

• Add $rint statements to function "y

leaing off

• key"oard

• de"ug and "reak$oint



Conclusions

• Buick tour of matla"- you should teach

yourself the rest. e*ll gie hints in

$ro"lem sets.• (inear alge"ra allows geometric

mani$ulation of $oints.

• (earn to loe !3?.

mat lab tutorial and linear algebra review

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