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.
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
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,
set of linear
equations is easy in Matlab. It is, maybe, the most used
operation in science and engineering, too.
Solving a system of
on a computer is nowadays as basic as doing arithmetic
a calculator. Let's see how easy Matlab makes this task...
- Circuit Analysis
One important algebra
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.
example, let's examine the following electrical circuit (resistors are
in ohms, currents in amperes, and voltages are in volts)...
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...
In Matlab there are several built-in functions provided for matrix
factorization (also called decomposition).
The name of the built-in
function for a Lower-Upper decomposition is 'lu'. To get the LU
factorization of a square matrix A, type the command '[L, U] = lu(A)'...
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
S V' (the apostrophe after a matrix or vector
means its transpose)
is an orthogonal m
is an r x r diagonal matrix,
is an n x n square orthogonal
matrix. Since U
are orthogonal, then...
Modified Gauss-Jordan Algorithm
A modified version of the classical method for performing the inversion without explicitly
generating the unit matrix by replicating its functionality within the
original matrix space for more efficient utilization of computational
resources is presented in this article.
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