Announcements
• Class mailing list: send email to
Hyoungjune Yi: aster@cs.umd.edu
• Homework at the end of class.
• Text is on reserve in the CS library.
• Powerpoint should be available by
10am class day.

Matlab tutorial and Linear
Algebra Review
• Today’s goals:
• Learn enough matlab to get started.
• Review some basics of Linear Algebra
• Essential for geometry of points and lines.
• But also, all math is linear algebra.
• (ok slight exaggeration).
• Many slides today adapted from Octavia
Camps, Penn State.

Starting Matlab
• For PCs, Matlab should be a program.
• For Sun’s:
Numerical
Analysis and Visualization
Matlab 6.1

Help
• help
• help command
Eg., help plus
• Help on toolbar
• demo
• Tutorial: http://amath.colorado.edu/scico/tutorials
/matlab/

Matlab interpreter
• Many common functions: see help ops

Vectors
• Ordered set of numbers: (1,2,3,4)
• Example: ( x,y,z ) v v

 coordinates of pt in space.
If v
( x
1
, x
2
,  , x n
)
 i n


1 x i
2
1 , v is a unit vecto r

Indexing into vectors

Vector Addition v
 w

( x
1
, x
2
)

( y
1
, y
2
)

( x
1
 y
1
, x
2
 y
2
) v
V+w w

Scalar Product a v
 a ( x
1
, x
2
)

( ax
1
, ax
2
) av v

Operations on vectors
• sum
• max, min, mean, sort, …
• Pointwise: .^

Inner (dot) Product v
 w v .
w

( x
1
, x
2
).( y
1
, y
2
)
 x
1 y
1
 x
2
.
y
2
The inner product is a SCALAR!
v .
w

( x
1
, x
2
).( y
1
, y
2
)

|| v ||

|| w || cos
 v .
w

0
 v
 w

A n
 m








 a a a
31 a

11
21 n 1 a
12 a
22 a
32
 a n 2




 a a a

1 m
2
3 m m a nm








Matrices
Sum:
C n
 m

A n
 m

B n
 m c ij
 a ij
 b ij
A and B must have the same dimensions

Matrices
Product:
C n
 c ij p


A n
 m k m 

1
B m
 a ik b kj p
A and B must have compatible dimensions
A n
 n
B n
 n

B n
 n
A n
 n
1
Identity Matrix:
I



0

0
0
1

0




0
0



1
IA

AI

A

Transpose:
C m
 n
 c
 ij
A
T n
 m a ji
Matrices
( A

B )
T 
A
T 
B
T
( AB )
T 
B
T
A
T
If A
T 
A A is symmetric

Matrices
Determinant: A must be square det

 a
11 a
21 a
12 a
22


 a
11 a
21 a
12 a
22
 a
11 a
22
 a
21 a
12 det

 a
11

 a a
21
31 a
12 a
22 a
32 a
13 a
23 a
33




 a
11 a
22 a
32 a
23 a
33
 a
12 a
21 a
31 a
23 a
33
 a
13 a
21 a
31 a
22 a
32

Matrices
Inverse:
A n
 n
A

1 n
 n

 a
11 a
21

A

1 n
 n
A n
 n

I a
12 a
22



1
 a
11 a
22
1
 a
21 a
12


 a a
22
21
A must be square
 a a
12


11

Indexing into matrices

Euclidean transformations

P
2D Translation
P’ t

t y y
2D Translation Equation x
P t
P’
P

( x , y ) t

( t x
, t y
) t x
P '

( x
 t x
, y
 t y
)

P
 t

t y y
2D Translation using Matrices x
P t
P’ t x
P '


 x y
 t x
 t y


P

( x , y ) t

( t x
, t y
)



1
0
0
1 t x t y t






 x y
1




P

P
P’
Scaling

s.y
y x
P
Scaling Equation
P’ s.x
P

( x , y )
P '

( sx , sy )
P '
 s

P
P '


 sx sy




 s
0
P '

S

P
S
0 s




 x y



P’
P
Rotation

Y’ y
Rotation Equations
Counter-clockwise rotation by an angle 
P’
X’ x

P

 x y
'
'




 cos sin


P'

R.P
 sin cos

 


 x y



Degrees of Freedom

 x y
'
'




 cos sin


 sin cos

 


 x y


R is 2x2 4 elements
BUT! There is only 1 degree of freedom: 
The 4 elements must satisfy the following constraints:
R

R
T 
R
T det( R )

1

R

I

S y
.y
y x
P
Stretching Equation
P

( x , y )
P’
P '

( s x x , s y y )
S x
.x
P '



 s s x y x y






 s x
0
S
0 s y




 x y


P '

S

P

Stretching = tilting and projecting
(with weak perspective)
P '



 s s x y x y






 s
0 x
0 s y




 x y


 s y



 s s y
0 x 0
1




 
 x y



Linear Transformation
P '


 a c



 sin
 cos
 b d




 x y

 cos
 sin



 
 s x
0
SVD
0 s y


 

 sin cos


 s y


 sin cos

 cos
 sin
 
 

 
 s x s y
0
0
1 




 

 sin cos

 cos
 sin





 x y

 cos sin





 x y



Affine Transformation
P '


 a c b d tx
 ty 




 x y
1





Files

Functions
• Format: function o = test(x,y)
• Name function and file the same.
• Only first function in file is visible outside the file.

Images

Debugging
• Add print statements to function by leaving off ;
• keyboard
• debug and breakpoint

Conclusions
• Quick tour of matlab, you should teach yourself the rest. We’ll give hints in problem sets.
• Linear algebra allows geometric manipulation of points.
• Learn to love SVD.