Download Linear Algebra - Welcome to the University of Delaware

Linear Algebra
A gentle introduction
Linear Algebra has become as basic and as applicable
as calculus, and fortunately it is easier.
--Gilbert Strang, MIT
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
1
: “shiv rpi”
What is a Vector ?

Think of a vector as a directed line
segment in N-dimensions! (has “length”
and “direction”)

Basic idea: convert geometry in higher
dimensions into algebra!
 Once you define a “nice” basis along
each dimension: x-, y-, z-axis …
 Vector becomes a 1 x N matrix!
 v = [a b c]T
 Geometry starts to become linear
algebra on vectors like v!
a 
  
v  b 
 c 
y
v
x
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
2
: “shiv rpi”
Vector Addition: A+B
vA+B
 w  ( x1 , x2 )  ( y1 , y2 )  ( x1  y1 , x2  y2 )
A
A+B = C
(use the head-to-tail method
to combine vectors)
B
C
B
A
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
3
: “shiv rpi”
Scalar Product: av
av  a( x1 , x2 )  (ax1 , ax2 )
av
v
Change only the length (“scaling”), but keep direction fixed.
Sneak peek: matrix operation (Av) can change length,
direction and also dimensionality!
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
4
: “shiv rpi”
Vectors: Dot Product
d 
A  B  AT B   a b c   e   ad  be  cf
 f 
The magnitude is the dot
product of a vector with itself
A  AT A  aa  bb  cc
2
A  B  A B cos( )
Think of the dot product as
a matrix multiplication
The dot product is also related to the
angle between the two vectors
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
5
: “shiv rpi”
Inner (dot) Product: v.w or wTv
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
If vectors v, w are “columns”, then dot product is wTv
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
6
: “shiv rpi”
Bases & Orthonormal Bases

Basis (or axes): frame of reference
vs
Basis: a space is totally defined by a set of vectors – any point is a linear
combination of the basis
Ortho-Normal: orthogonal + normal
[Sneak peek:
Orthogonal: dot product is zero
Normal: magnitude is one ]
Rensselaer Polytechnic Institute
7
x  1 0 0
y  0 1 0
x y  0
xz  0
z  0 0 1
yz  0
T
T
T
Shivkumar Kalyanaraman
: “shiv rpi”
What is a Matrix?

A matrix is a set of elements, organized into rows and
columns
rows
columns
a b 
c d 


Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
8
: “shiv rpi”
Basic Matrix Operations

Addition, Subtraction, Multiplication: creating new matrices (or functions)
a b   e
c d    g

 
f  a  e b  f 


h  c  g d  h 
a b   e
c d    g

 
f  a  e b  f 



h  c  g d  h 
a b   e
c d   g


f  ae  bg


h  ce  dg
af  bh
cf  dh 
Just add elements
Just subtract elements
Multiply each row
by each column
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
9
: “shiv rpi”
Matrix Times Matrix
L  MN
l11 l12 l13   m11 m12
l


 21 l22 l23   m21 m22
l31 l32 l33  m31 m32
m13   n11 n12


m23   n21 n22
m33  n31 n32
n13 

n23 
n33 
l12  m11n12  m12n22  m13n32
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
10
: “shiv rpi”
Multiplication

Is AB = BA? Maybe, but maybe not!
a b   e
c d   g





f  ae  bg ...


h   ...
...
e
g

f  a b  ea  fc ...




h   c d   ...
...
Matrix multiplication AB: apply transformation B first, and
then again transform using A!
Heads up: multiplication is NOT commutative!
Note: If A and B both represent either pure “rotation” or
“scaling” they can be interchanged (i.e. AB = BA)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
11
: “shiv rpi”
Matrix operating on vectors





Matrix is like a function that transforms the vectors on a plane
Matrix operating on a general point => transforms x- and y-components
System of linear equations: matrix is just the bunch of coeffs !
x’ = ax + by
y’ = cx + dy
a b   x   x'

    
c d   y  y'
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
12
: “shiv rpi”
Direction Vector Dot Matrix
 ax
a
v  M  v   y
 az

0
bx
by
bz
0
cx
cy
cz
0
d x   vx 
d y  v y 

d z   vz 
  
1  1
vx  vx ax  v y bx  vz cx
v  v x a  v y b  v z c
vy  vx a y  v y by  vz c y
vz  vx az  v y bz  vz cz
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
13
: “shiv rpi”
Matrices: Scaling, Rotation, Identity



Pure scaling, no rotation => “diagonal matrix” (note: x-, y-axes could be scaled differently!)
Pure rotation, no stretching => “orthogonal matrix” O
Identity (“do nothing”) matrix = unit scaling, no rotation!
r1 0
0 r2
[0,1]T
[0,r2]T
scaling
[r1,0]T
[1,0]T
cos -sin
sin cos
[0,1]T
rotation
[-sin, cos]T
[cos, sin]T

[1,0]T
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
14
: “shiv rpi”
Scaling
P’
P
a.k.a: dilation (r >1),
contraction (r <1)
r 0
0 r
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
15
: “shiv rpi”
Rotation
P
P’
cos -sin
sin cos
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
16
: “shiv rpi”
2D Translation
P’
t
P
P'  ( x  t x , y  t y )  Pt
ty
y
P
x
Rensselaer Polytechnic Institute
17
P’
t
tx
Shivkumar Kalyanaraman
: “shiv rpi”
Inverse of a Matrix




Identity matrix:
AI = A
Inverse exists only for square
matrices that are non-singular
 Maps N-d space to another
N-d space bijectively
Some matrices have an
inverse, such that:
AA-1 = I
Inversion is tricky:
(ABC)-1 = C-1B-1A-1
Derived from noncommutativity property
1 0 0


I  0 1 0 
0 0 1
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
18
: “shiv rpi”
Determinant of a Matrix


Used for inversion
If det(A) = 0, then A has no inverse
a b 
A

c
d


det( A)  ad  bc
1  d  b
A 
ad  bc  c a 
1
http://www.euclideanspace.com/maths/algebra/matrix/functio
ns/inverse/threeD/index.htm
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
19
: “shiv rpi”
Projection: Using Inner Products (I)
p = a (aTx)
||a|| = aTa = 1
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
20
: “shiv rpi”
Homogeneous Coordinates
Represent coordinates as (x,y,h)
 Actual coordinates drawn will be (x/h,y/h)

Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
21
: “shiv rpi”
Homogeneous Coordinates

The transformation matrices become 3x3 matrices,
and we have a translation matrix!
x’
y’ =
1
New point
1
0
0
0
1
0
tx
ty
1
x
y
1
Transformation
Exercise: Try composite translation.
Rensselaer Polytechnic Institute
22
Original point
Shivkumar Kalyanaraman
: “shiv rpi”
Homogeneous Transformations
v  M  v
vx   a1
v    a
 y   2
 vz   a3
  
1 0
b1
c1
b2
b3
0
c2
c3
0
d1  v x 
d 2  v y 

d3  vz 
  
1  1
vx  a1v x  b1v y  c1v z  d1
vy  a2 v x  b2 v y  c2 v z  d 2
vz  a3v x  b3v y  c3v z  d 3
1  0v x  0v y  0v z  1
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
23
: “shiv rpi”
Order of Transformations
Note that matrix on the right is the first applied
 Mathematically, the following are equivalent
p’ = ABCp = A(B(Cp))
 Note many references use column matrices to
represent points. In terms of column matrices
p’T = pTCTBTAT

T
R
M
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
24
: “shiv2rpi”
Rotation About a Fixed Point other than
the Origin
Move fixed point to origin
Rotate
Move fixed point back
M = T(pf) R() T(-pf)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
25
: “shiv2rpi”
Vectors: Cross Product

The cross product of vectors A and B is a vector C which is
perpendicular to A and B

The magnitude of C is proportional to the sin of the angle
between A and B

The direction of C follows the right hand rule if we are
working in a right-handed coordinate system
A  B  A B sin(  )
A×B
B
A
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
26
: “shiv rpi”
MAGNITUDE OF THE CROSS
PRODUCT
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
27
: “shiv rpi”
DIRECTION OF THE CROSS
PRODUCT

The right hand rule determines the direction of the
cross product
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
28
: “shiv rpi”
For more details


Prof. Gilbert Strang’s course videos:
http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring2005/VideoLectures/index.htm

Esp. the lectures on eigenvalues/eigenvectors, singular value
decomposition & applications of both. (second half of course)

Online Linear Algebra Tutorials:
http://tutorial.math.lamar.edu/AllBrowsers/2318/2318.asp

Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
29
: “shiv rpi”