Matlab tutorial and Linear
Algebra Review
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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
v  ( x1 , x2 ,  , xn )
• Ordered set of
numbers: (1,2,3,4)
v
• Example: (x,y,z)
coordinates of pt in
If
space.

n
2
i 1
i
x
v  1, v is a unit vecto r
Indexing into vectors
Vector Addition
v  w  ( x1 , x2 )  ( y1 , y2 )  ( x1  y1 , x2  y2 )
v
V+w
w
Scalar Product
av  a( x1 , x2 )  (ax1 , ax2 )
av
v
Operations on vectors
• sum
• max, min, mean, sort, …
• Pointwise: .^
Inner (dot) Product
v

w
v.w  ( x1 , x2 ).( y1 , y2 )  x1 y1  x2 . y2
The inner product is a SCALAR!
v.w  ( x1 , x2 ).( y1 , y2 ) || v ||  || w || cos
v.w  0  v  w
Matrices
Anm
 a11 a12
a
 21 a22
 a31 a32


 
an1 an 2
 a1m 
 a2 m 
 a3m 

  
 anm 
Sum:
Cnm  Anm  Bnm
cij  aij  bij
A and B must have the same
dimensions
Matrices
Product:
Cn p  Anm Bm p
m
cij   aik bkj
k 1
Identity Matrix:
A and B must have
compatible dimensions
Ann Bnn  Bnn Ann
1 0  0


0 1  0

I 
IA

AI

A
   


0 0  1
Matrices
Transpose:
Cmn  A nm
cij  a ji
T
If
AT  A
( A  B)  A  B
T
T
( AB)  B A
T
A is symmetric
T
T
T
Matrices
Determinant:
A must be square
 a11 a12  a11 a12
det 

 a11a22  a21a12

a21 a22  a21 a22
 a11 a12
det a21 a22
a31 a32
a13 
a22

a23   a11
a32

a33 
a23
a33
 a12
a21 a23
a31 a33
 a13
a21 a22
a31 a32
Matrices
Inverse:
Ann A
A must be square
1
nn
A
1
1
nn
Ann  I
 a11 a12 
1

a

a
a11a22  a21a12
21
22


 a22  a12 
 a

a
21
11


Indexing into matrices
Euclidean transformations
2D Translation
P’
t
P
2D Translation Equation
ty
P
P’
t
y
x
P  ( x, y )
t  (t x , t y )
tx
P'  ( x  t x , y  t y )  Pt
2D Translation using Matrices
ty
P
P’
t
y
x
tx
P  ( x, y )
t  (t x , t y )
t
 x
 x  t x  1 0 t x   
P'  

  y


 y  t y  0 1 t y   1 
 
P
Scaling
P’
P
Scaling Equation
P’
s.y
P
y
P  ( x, y )
P'  ( sx, sy )
P'  s  P
x
s.x
 sx   s 0  x 
P'     
 

 sy  0 s   y 
P'  S  P
S
Rotation
P
P’
Rotation Equations
Counter-clockwise rotation by an angle 
Y’
 x' cos
 y '   sin 
  
P’

P
y
X’
x
P'  R.P
 sin    x 



cos   y 
Degrees of Freedom
 x' cos
 y '   sin 
  
R is 2x2
 sin    x 



cos   y 
4 elements
BUT! There is only 1 degree of freedom: 
The 4 elements must satisfy the following constraints:
R  RT  RT  R  I
det( R )  1
Stretching Equation
P  ( x, y )
P '  ( s x x, s y y )
P’
Sy.y
 sx x  sx
P'     
s
y
0
 y  
P
y
x
Sx.x
0   x
 

sy   y
S
P'  S  P
Stretching = tilting and projecting
(with weak perspective)
 sx x  sx
P'     
s
y
0
y
  
sx

0   x
s


s
s y   y  y  y
 0

0  x 
 
  y
1
Linear Transformation
a b   x 
P'  
 

c d   y
 sin 

 cos
 sin 
 sy 
 cos
SVD
cos   s x
0

sin   
 sx
cos  
sy

sin   
 0
0  sin 
s y  cos 

0  sin 

  cos 
1
cos    x 
 

sin    y 
cos    x 
sin    y 
Affine Transformation
 x
a b tx  
P'  

y

 
c
d
ty

 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.