Linear Algebra - (Linear system|System of Linear equations)

> Linear Algebra

1 - About

In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.

<MATH> \begin{alignat}{7} 3x &&\; + \;&& 2y &&\; - \;&& z &&\; = \;&& 1 & \\ 2x &&\; - \;&& 2y &&\; + \;&& 4z &&\; = \;&& -2 & \\ -x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z &&\; = \;&& 0 & \end{alignat} </MATH>

Each linear system corresponds to a linear system with zero right-hand sides:

  • <math>a_1 . x = beta_1 \rightarrow a_1 . x = 0</math>
  • <math>a_2 . x = beta_2 \rightarrow a_2 . x = 0</math>
  • <math>a_n . x = beta_n \rightarrow a_n . x = 0</math>

If a linear system has a solution u1 then that solution is unique if the only solution to the corresponding homogeneous linear system is 0.

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3 - Type

3.1 - Homogeneous

A system of homogeneous linear equations is called a homogeneous linear system.

<MATH> \begin{alignat}{7} 3x &&\; + \;&& 2y &&\; - \;&& z &&\; = \;&& 0 & \\ 2x &&\; - \;&& 2y &&\; + \;&& 4z &&\; = \;&& 0 & \\ -x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z &&\; = \;&& 0 & \end{alignat} </MATH>

The solution set of a homogeneous linear system is a vector space.

Lemma: Let u1 be a solution to a linear system. Then, for any other vector u2, u2 is also a solution if and only if u2 - u1 is a solution to the corresponding homogeneous linear system

3.2 - Triangular

A triangular linear system as a triangular form

  • <math>10 = 2x_1 + 3x_2 - 4x_3</math>
  • <math>~3 = 1x_2 + 2x_3</math>
  • <math>15 = 5x_3</math>

and can be expressed as a triangular matrix

4 - Documentation / Reference

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linear_algebra/linear_system.txt · Last modified: 2013/08/21 16:39 by gerardnico