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Dan Witzner Hansen Email : [email protected] Linear algbra. Last week?. Groups? Improvements – what is missing?. Misc. The goal is to be able to solve linear equations Continue with linear algebra Linear mappings Basis vectors & independence Solving linear equations & Determinants

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Dan witzner hansen email witzner@itu dk

Dan Witzner Hansen

Email:

[email protected]

Linear algbra



Groups?

Improvements – what is missing?

Misc


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

Today



What is a linear equation
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
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
Matrix Form of Linear System

Compact notation Ax=b


Linear mappings
Linear Mappings

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?

Example

Species



Matlab code
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 clarity

y2 = (-A(2,1)*x+b(2))/A(2,2);

Plot(x,y1,'r-','LineWidth',3); hold on

Plot(x,y2,'b-','LineWidth',3); hold off

title('Linear equations and their solution')



An now for some formalism

Subspaces

Independent vectors

Basis vectors / Orthonomal basis

An now for some formalism


Subspaces

A subspaceis a vector space contained in another vector space

Subspaces


Independent vectors
Independent vectors

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
Basis vectors - example

Change of basis


Show that it is an orthonormal basis
Show that it is an Orthonormal basis


What happens with this one
What happens with this one?

A = [1 4 2;2 8 6; 3 124];

[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')






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.

Solutions of linear equations


Recall
Recall

Solution


Solving systems algebraically
Solving systems algebraically

Which 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

Determinant

>>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?

What if?


Matrix inverse

Solve simple linear equation

Matrix inverse:

A (unique) inverse exist if det(A) ≠ 0 (NxN matrices)

Matlab: >>invA =inv(A)

Matrix Inverse


Solving linear systems
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 Ais not invertible

  • If n>m then under constrained.



Notice implementation

Don’tuse for solvingthe linear system. It is mostlymeant for notationalconvenience.

It is faster and more accurate (numerically) to write (solve)x=A\bthaninv(A)*b:

Notice: implementation


Simple inversion of some matrices

Diagonal matrices

Orthogonal matrices

Simple inversion of (some) matrices


Fitting lines
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:


Example fitting a line
Example: Fitting a Line

  • Suppose we have points (2, 1), (5, 2), (7, 3), and (8, 3)?????


Fitting lines1

With more than two points, there is no guarantee that they will all be on the same line

Fitting Lines

courtesy of

Vanderbilt U.


Least squares
Least squares

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 lines2
Fitting Lines

Solution: Use the pseudoinverse

A+ =(ATA)-1AT to obtain least-squares solutionx=A+b

courtesy of

Vanderbilt U.


Example fitting a line1

and x=A+b=(0.3571, 0.2857)T

Example: Fitting a Line

  • Suppose we have points (2, 1), (5, 2), (7, 3), and (8, 3)

  • Then????


Example fitting a line2
Example: Fitting a Line

(2, 1), (5, 2), (7, 3), and (8, 3)


Homogeneous systems of equations
Homogeneous Systems of Equations

  • Suppose we want to solve Ax = 0

  • There is a trivial solution x = 0, but we don’t want this. For what other values of x is Ax 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)




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.

Properties of SVD


Example line fitting as a homogeneous system
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
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 tox =(0.3571, 0.2857)T


Robust methods
Robust methods

  • So what about outliers

  • Other metrics such as other norms

  • More about this later



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