**Outline**

What is a linear system?

Examples of systems arising in applications

**Systems of equations**

A general category of numerical problem is systems of
equations in

which we have m statements about n variables x_{1}, . . . , x_{n}:

y_{1} = f_{1}(x_{1}, . . . , x_{n})

y_{2} = f_{2}(x_{1}, . . . , x_{n})

...

y_{m} = f_{m}(x_{1}, . . . , x_{n})

The fs are known functions, the ys are given quantities,
and the xs

are the unknown variables to be determined.

**Linear functions
**A function f is linear if

f(x + x') = f(x) + f(x')

and

f(ax) = af(x)

for any a, x and x'.

If f is a function of one variable, it has to be f(x) = ax for some a.

If f is a function of n variables it has the form

f(x1, . . . , x_{n}) = a_{1}x_{1} + a_{2}x_{2} + · · · + a_{n}x_{n}.

**Systems of linear equations
**A linear system is just a system of equations

y_{i} = f_{i}(x_{1}, . . . , x_{n})

where the functions f_{i} that define it are linear. This means a system

of m linear equations in n variables looks like this:

(I have switched to the convention for linear systems that the un-

knowns go on the right.)

**Matrix form of linear system**

A linear system can be written as one equation using matrix notation.

A small example:

becomes

Ax = b

For those who enjoy dots:

Ax = b

A system of m equations in n unknowns is an equality with an

m-vector on one side and the product of an m × n matrix with an

n-vector on the other side.

**Geometric transformations**

Suppose we have a coordinate system transformation:

x' = ax + by

y' = cx + dy

and we want to nd the (x, y) that goes to (x', y'). This is a 2 × 2

linear system.

**Circuit analysis**

Here is a simple circuit:

What are the voltages across all the components? A straightforward

analysis but tedious on paper.

• Variables

◦
voltages at nodes: v_{a}, . . . , v_{e}

◦
currents through voltage sources I_{1}, I_{2}

• Equations

◦
net current at each node is zero

◦
known voltages across voltage sources

Result is a system with 7 variables and 7 equations

**Radiative transfer**

Problem in heat transfer: equilibrium energy distribution.

In, for example, a furnace, surfaces exchange heat by thermal

radiation.

• Each surface emits radiation equally in all directions at some rate

• Each surface reflects a fraction of the incident radiation

• The emitted radiation from one surface falls on the other

surfaces

◦
How it gets distributed depends on geometry

◦
The fraction of light leaving patch i that ends up at patch j is the

form factor f_{ij} .

A radiation balance at each surface results in a linear equation:

where B_{i} is the total radiation leaving the surface, known as its

radiosity (hence the name of the method).