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Linear Systems I


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 x1, . . . , xn:

y1 = f1(x1, . . . , xn)
y2 = f2(x1, . . . , xn)
ym = fm(x1, . . . , xn)

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')


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, . . . , xn) = a1x1 + a2x2 + · · · + anxn.

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

yi = fi(x1, . . . , xn)

where the functions fi 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:


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: va, . . . , ve
◦ currents through voltage sources I1, I2

• 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

• 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
◦ How it gets distributed depends on geometry
◦ The fraction of light leaving patch i that ends up at patch j is the
form factor fij .

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

where Bi is the total radiation leaving the surface, known as its
radiosity (hence the name of the method).