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- 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

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 Ax=b

Linear Mappings

Affine mapping

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?

ExampleSpecies

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')

A subspaceis a vector space contained in another vector space

SubspacesIndependent 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

Change of basis

Show that it is an Orthonormal basis

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')

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 equationsRecall

Solution

Solving systems algebraically

Which solution(s)?

Can we always do this?

How many solutions are there?

- 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

>>det(A)

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:

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

Matlab: >>invA =inv(A)

Matrix InverseSolving 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.

Solve Ax=b (notice multiply from right):

Matrix inverse exampleDon’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: implementationFitting 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

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

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

Fitting Linescourtesy of

Vanderbilt U.

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 Lines

Solution: Use the pseudoinverse

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

courtesy of

Vanderbilt U.

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 Line

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

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)

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 SVDexample: 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 tox =(0.3571, 0.2857)T

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