Linear Algebra - Welcome to the University of Delaware
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Transcript 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
T
y 0 1 0
T
z 0 0 1
T
x y 0
x z 0
yz 0
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 af bh
h ce dg 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 MN
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
cx
by
cy
bz
cz
0
0
d x vx
d y v y
d z vz
1 1
vx vx ax v y bx vz cx
v v x a v y b v z c
vy vx a y v y by vz c y
vz 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 ) Pt
ty
y
P
x
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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.
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Original point
Shivkumar Kalyanaraman
: “shiv rpi”
Homogeneous Transformations
v M v
vx a1
v a
y 2
vz a3
1 0
b1
c1
b2
c2
b3
0
c3
0
d1 v x
d 2 v y
d3 vz
1 1
vx a1v x b1v y c1v z d1
vy a2 v x b2 v y c2 v z d 2
vz 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”