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LAST WEEK?
mnm
n
aa
aaa
A
1
11111
h d g c
f b e a
h g
f e
d c
b a
h d g c
f b e a
h g
f e
d c
b a
dh cf dg ce
bh af bg ae
h g
f e
d c
b a
MISC
Groups?
Improvements – what is missing?
TODAY
The goal is to be able to solve linear equations
Continue with linear algebra
Linear mappings
Basis vectors & independence
Solving linear equations & Determinants
Inverse & Least squares
SVD
Lot’s of stuff. Don’t despair – you will be greatly rewarded in the future
LINEAR SYSTEMS
WHAT IS A LINEAR EQUATION?
A linear equation is an equation of the form,
anxn+ an-1xn-1+ . . . + a1x1 = b.
WHAT IS A SYSTEM OF LINEAR EQUATIONS?
A system of linear equations is simply a set of linear equations. i.e.
a1,1x1+ a1,2x2+ . . . + a1,nxn = b1
a2,1x1+ a2,2x2+ . . . + a2,nxn = b2
. . .
am,1x1+ am,2x2+ . . . + am,nxn = bm
MATRIX FORM OF LINEAR SYSTEM
Compact notation A x = b
LINEAR MAPPINGS
A
Axy baxy
bAxy Affine mapping
EXAMPLE
Species 1: eats 5 units of A and 3 of B. Species 2: eats 2 units of A and 4 of B. Everyday a total of 900 units of A and
960 units of B are eaten. How many animals of each species are there?
€
5x1 + 2x2 = 900
3x1 + 4x2 = 960
Species
EXAMPLE:
€
5 2
3 4
⎡
⎣ ⎢
⎤
⎦ ⎥x1
x2
⎡
⎣ ⎢
⎤
⎦ ⎥=
900
960
⎡
⎣ ⎢
⎤
⎦ ⎥
Ax = b
€
5x1 + 2x2 = 900
3x1 + 4x2 = 960
MATLAB CODE
A = [5 2; 3 4];b = [900 960];
x = linspace(0,150,100);y1 = (-A(1,1)*x+b(1))/A(1,2); %made for clarityy2 = (-A(2,1)*x+b(2))/A(2,2);
Plot(x,y1,'r-','LineWidth',3); hold onPlot(x,y2,'b-','LineWidth',3); hold offtitle('Linear equations and their solution')
BASIS, INDEPENDENCE AND SUBSPACES
AN NOW FOR SOME FORMALISM
Subspaces Independent vectors Basis vectors / Orthonomal basis
SUBSPACES
A subspace is a vector space contained in another vector space
INDEPENDENT VECTORS
Axy Can it happen that y=0 if x is nonzero?
If y is non-zero for all non-zero x, then the column vectors of A are said to be linear independent.
These vectors form a set of basis vectors
Orthonormal basis when the vectors are unit size and orthogonal.
BASIS VECTORS - EXAMPLE
0
0
1
1e
0
1
0
2e
1
0
0
3e
4
3
2
1
0
0
4
0
1
0
3
0
0
1
2x
€
x =
2
3
4
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥=
1 0 0
0 1 0
0 0 1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
2
3
4
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Change of basis
SHOW THAT IT IS AN ORTHONORMAL BASIS
€
b1 =cosv
sinv
⎡
⎣ ⎢
⎤
⎦ ⎥,b2 =
−sinv
cosv
⎡
⎣ ⎢
⎤
⎦ ⎥
vv
vvB
cossin
sincos
WHAT HAPPENS WITH THIS ONE?
4123
682
241
A = [1 4 2;2 8 6; 3 12 4];[X,Y,Z] = meshgrid(-10:10,-10:10,-10:10);x = [X(:),Y(:),Z(:)]’;p = A*x;plot3(p(1,:),p(2,:),p(3,:),'rx')
A CLOSER LOOK AT MATRIX MULTIPLICATION
nmnmm
nn
nn
xaxaxa
xaxaxa
xaxaxa
Ax
2211
2222121
1212111
mn
n
n
n
mm a
a
a
x
a
a
a
x
a
a
a
x
2
1
2
22
12
2
1
21
11
1
nnAxAxAx :,2:,21:,1
2D EXAMPLE
€
Ax =a11 a12
a21 a22
⎡
⎣ ⎢
⎤
⎦ ⎥x1
x2
⎡
⎣ ⎢
⎤
⎦ ⎥
€
Ax =a11 a12
a21 a22
⎡
⎣ ⎢
⎤
⎦ ⎥x1
x2
⎡
⎣ ⎢
⎤
⎦ ⎥=x1
a11
a21
⎡
⎣ ⎢
⎤
⎦ ⎥+ x2
a12
a22
⎡
⎣ ⎢
⎤
⎦ ⎥
WHAT HAPPENS WITH THIS ONE?
4123
682
241
4x =
SOLVING LINEAR EQUATIONS
SOLUTIONS OF LINEAR EQUATIONS
A solution to a system of equations is simply an assignment of values to the variables that satisfies (is a solution to) all of the equations in the system.
If a system of equations has at least one solution, we say it is consistent.
If a system does not have any solutions we say that it is inconsistent.
RECALL
€
5 2
3 4
⎡
⎣ ⎢
⎤
⎦ ⎥x1
x2
⎡
⎣ ⎢
⎤
⎦ ⎥=
900
960
⎡
⎣ ⎢
⎤
⎦ ⎥
Ax = b€
5x1 + 2x2 = 900
3x1 + 4x2 = 960
Solution
SOLVING SYSTEMS ALGEBRAICALLY
2x 3y z 5
y z 1
z 3Which solution(s)?
Can we always do this?
How many solutions are there?
DETERMINANT
For A (2x2 matrix)
When det A ≠0 a unique solution exists (nonsingular) When det A =0 the matrix is singular (lines same
slope) and are therefore the columns are linear dependent Coincident (infinitely many solutions) Parallel (no solutions)
Determinant can be used when solving linear equations (Cramers’ rule), but not useful in practice
12212211det aaaaA >>det(A)
WHAT IF?
What to do when the dimension and the number of data points is large?
How many data points are needed to solve for the unknown parameters in x?
MATRIX INVERSE
Solve simple linear equation
Matrix inverse:
A (unique) inverse exist if det(A) ≠ 0 (NxN matrices)
Matlab: >>invA =inv(A)
€
ax = b
a−1ax = a−1bx
x = a−1b
0000
000
0010
0001
1
IAA
SOLVING LINEAR SYSTEMS
If m = n (A is a square matrix & Det(A)!=0), then we can obtain the solution by simple inversion (:
If m > n, then the system is over-constrained and A is not invertible
If n>m then under constrained.
MATRIX INVERSE EXAMPLE
Solve Ax=b (notice multiply from right):
6
4,
24
35
2
1,
54
32 1 bAA
bAx 1
NOTICE: IMPLEMENTATION
Don’t use for solving the linear system. It is mostly meant for notational convenience.
It is faster and more accurate (numerically) to write (solve) x=A\b than inv(A)*b:
bA 1
1111)( ABCABC
TT AA )()( 11
SIMPLE INVERSION OF (SOME) MATRICES Diagonal matrices
Orthogonal matrices
nnd
d
d
D
000
000
000
000
22
11
T
T
AA
IAA
1
,cossin
sincos,
cossin
sincos 1
TRRR
1
122
111
1
000
000
000
000
nnd
d
d
D
FITTING LINES
A 2-D point x = (x, y) is on a line with
slope m and intercept b if and only if y =
mx + b Equivalently,
So the line defined by two points x1, x2 is
the solution to the following system of equations:€
x 1[ ]m
b
⎡
⎣ ⎢
⎤
⎦ ⎥= y
€
x1 1
x2 1
⎡
⎣ ⎢
⎤
⎦ ⎥m
b
⎡
⎣ ⎢
⎤
⎦ ⎥=y1
y2
⎡
⎣ ⎢
⎤
⎦ ⎥
EXAMPLE: FITTING A LINE
Suppose we have points (2, 1), (5, 2), (7, 3), and (8, 3)?????
3
3
2
1
18
17
15
12
b
m
FITTING LINES
With more than two points, there is no guarantee that they will all be on the same line
courtesy ofVanderbilt U.
LEAST SQUARESFxy
)(,minargˆ Fxyrrx Objective:
Find the vector Fx in the column range of F, which is closest to the right-hand side vector y.The residual r=y-Fx
FITTING LINES
courtesy ofVanderbilt U.
Solution: Use the pseudoinverse
A+ = (ATA)-1AT to obtain least-
squares solution x = A+b
EXAMPLE: FITTING A LINE
Suppose we have points (2, 1), (5, 2), (7, 3), and (8, 3)
Then????
and x = A+b = (0.3571, 0.2857)T
EXAMPLE: FITTING A LINE(2, 1), (5, 2), (7, 3), and (8, 3)
HOMOGENEOUS SYSTEMS OF EQUATIONS
Suppose we want to solve A x = 0 There is a trivial solution x = 0, but we
don’t want this. For what other values of x is A x close to 0?
This is satisfied by computing the singular value decomposition (SVD) A = UDVT (a non-negative diagonal matrix between two orthogonal matrices) and taking x as the last column of V In Matlab [U, D, V] = svd(A)
SINGULAR VALUE DECOMPOSITION
Tnnnmmmnm VΣUA
IUU T
021 n
IVV T
nm
XXVT XVΣ T XVUΣ T
TTTnnn VUVUVUA 222111
000
00
00
00
Σ
2
1
n
NULL SPACE & IMAGE SPACE
4123
682
241
PROPERTIES OF SVD
When the columns of A =UDV are independent then all
Tells how close to singular A is. Inverse and pseudoinverse The columns of U corresponding to
nonzeros singular values span the range of A, the columns of V corresponding to zero singular values the nullspace.
€
dii > 0
0000
0000
000
000
Σ 2
1
TUVDA 11 TUVDA 1
0
EXAMPLE: LINE-FITTING AS A HOMOGENEOUS SYSTEM
A 2-D homogeneous point x = (x, y, 1)T is on the line l = (a, b, c)T only when
ax + by + c = 0
We can write this equation with a dot product:
x .l = 0, and hence the following system is implied for
multiple points x1, x2, ..., xn:
EXAMPLE: HOMOGENEOUS LINE-FITTING
Again we have 4 points, but now in homogeneous form:
(2, 1, 1), (5, 2, 1), (7, 3, 1), and (8, 3, 1)
The system of equations is:
Taking the SVD of A, we get:compare to x = (0.3571, 0.2857)T
ROBUST METHODS
So what about outliers
• Other metrics such as other norms• More about this later
NEXT WEEK
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