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Linear
Algebra
Here are some important and requested topics regarding Linear Algebra. This
type of algebra is a branch of mathematics related to the
study of vectors (families of vectors or linear spaces), and
with functions that enter one vector and produce another, according to
certain rules. These functions are called linear maps and are
usually represented by matrices. Linear algebra is quite relevant
in modern math and its applications.
Determinants
The
symbol which consists of the four numbers a1, b1, a2, b2 arranged in
two rows and two columns is called a determinant of second order or
determinant of order two. The four numbers are called elements of the
determinant...
Cramer's
Rule
In
linear
algebra, the
method of solution of
systems of equations by determinants is called Cramer's Rule. This rule for linear equations in
3 unknowns is a method of solving by
determinants the following equations for x,
y, z...
Simultaneous
Equations
Solving a
set of linear
equations is easy in Matlab. It is, maybe, the most used
operation in science and engineering, too.
Solving a system of
simultaneous equations
on a computer is nowadays as basic as doing arithmetic
additions using
a calculator. Let's see how easy Matlab makes this task...

Application
 Circuit Analysis
One important algebra
application
is the resolution of electrical circuits.
We can describe this type of circuits with linear equations, and then
we can solve the linear
system using Matlab.
For
example, let's examine the following electrical circuit (resistors are
in ohms, currents in amperes, and voltages are in volts)... 
Linear
Programming (as optimization problem)
We will illustrate the method of linear programming by means of a
simple example giving a numerical
solution. We are going to formulate the problem as an
optimization issue, and we'll use the instruction 'fminsearch', which
is an always available function...
LU
Factorization
In Matlab there are several builtin functions provided for matrix
factorization (also called decomposition).
The name of the builtin
function for a LowerUpper decomposition is 'lu'. To get the LU
factorization of a square matrix A, type the command '[L, U] = lu(A)'...
Singular
Value Decomposition  SVD
Let's suppose that a matrix A
is singular. Then, let A
be a real m
x n matrix
of rank r. The Singular Value
Decomposition (svd) of A
is A
= U
S V' (the apostrophe after a matrix or vector
means its transpose)
where U
is an orthogonal m
x n
matrix, S
is an r x r diagonal matrix,
and V
is an n x n square orthogonal
matrix. Since U
and V
are orthogonal, then...
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