# 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 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)'...

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...

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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