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Matrix Size & Validation
Vector Algebra
What is Vector?
Vector Norm
Unit Vector
Vector Addition
Vector Subtraction
Vector Scalar Multiple
Vector Multiplication
Vector Inner Product
Vector Outer Product
Vector Cross Product
Vector Triple Cross Product
Vector Triple Dot Product
Scalar Triple Product
Orthogonal & Orthonormal Vector
Cos Angle of Vectors
Scalar and Vector Projection
Matrix Algebra
What is a matrix?
Special Matrices
Matrix One
Null Matrix
Matrix Diagonal Is Diagonal Matrix?
Identity Matrix
Matrix Determinant
Matrix Sum
Matrix Trace
Matrix Basic Operation
Is Equal Matrix?
Matrix Transpose
Matrix Addition
Matrix Subtraction
Matrix Multiplication
Matrix Scalar Multiple
Hadamard Product
Horizontal Concatenation
Vertical Concatenation
Elementary Row Operations
Matrix RREF
Finding inverse using RREF (Gauss-Jordan)
Finding Matrix Rank using RREF
Matrix Inverse
Is Singular Matrix?
Linear Transformation
Matrix Generalized Inverse
Solving System of Linear Equations
Linear combination, Span & Basis Vector
Linearly Dependent & Linearly Independent
Change of basis
Matrix Rank
Matrix Range
Matrix Nullity & Null Space
Eigen System
Matrix Eigen Value & Eigen Vector
Symmetric Matrix
Matrix Eigen Value & Eigen Vector for Symmetric Matrix
Similarity Transformation and Matrix Diagonalization
Matrix Power
Orthogonal Matrix
Spectral Decomposition
Singular Value Decomposition
Resources on Linear Algebra

Solving System Linear Equations

By Kardi Teknomo, PhD.
LinearAlgebra

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Linear equation is an equation in the form of Solving System Linear Equations
The Solving System Linear Equationsare the known constant.
The Solving System Linear Equationsare the Solving System Linear Equationsunknown variables. The problem in linear equation is to find the values of the unknown variables that satisfy the equation.


For example, whenSolving System Linear Equations, Solving System Linear Equationsand Solving System Linear Equationswe have linear equation Solving System Linear Equationsand the solution isSolving System Linear Equations.

We may also have several equations and several unknowns that we would like to find out. A linear system is a set of Solving System Linear Equationslinear equations each in Solving System Linear Equationsunknown. We can write a linear system as
Solving System Linear Equations

When we have many equations and many unknowns, it is easier to represent the linear system into matrix. We put the constant coefficients of the equations into a matrix, and then we can multiply with the unknown to obtain the constants. To transform the system of linear equations into matrix format, you need to reorder the equations according to the order of the unknowns Solving System Linear Equationsand put the coefficients of the unknowns into matrix coefficientsSolving System Linear Equations. The constants on the right hand side of the equation are written into vector constantsSolving System Linear Equations. The linear system above can be written as
Solving System Linear Equations

Thus, a linear system can be simplified into a matrix product
Solving System Linear Equations

A solution of the linear system is an ordered collection of Solving System Linear Equationsnumbers that satisfies the Solving System Linear Equationslinear equations, which can be written in short as a vector solutionSolving System Linear Equations.

Example:
Solve a linear system with three equations and three unknowns
Solving System Linear Equations
The linear system can be written as Solving System Linear Equationswhere matrix
Solving System Linear EquationsSolving System Linear Equationsand Solving System Linear Equations
The solution of the linear system is
Solving System Linear Equations

The interactive program below will help you to solve a system of linear equationsSolving System Linear Equations. To use the program, first you need to transform your system of linear equations into matrix format as explained in the example above. Your input is matrix coefficients Solving System Linear Equationsand vector constantsSolving System Linear Equations. Then you click “Solve Linear System Ax=b” button and the program will produce the vector solutionSolving System Linear Equations. Optionally, you can select your output is either in decimal or in rational format. When you click “Random Example” button, it will create random input matrix to provide you with more examples of linear system. Note that if the coefficient matrix is singular or nearly singular, you will get only the approximate solution in least square sense using generalized inverse such that the error is minimized Solving System Linear Equations .

matrix A                                                         vector b

Report in rational format

Notes

Some important notes on linear systems are:

  • A linear system Solving System Linear Equationsis called non-homogeneous system when vector Solving System Linear Equationsis not a zero vector. A linear system Solving System Linear Equationsis called homogeneous system because vector Solving System Linear Equationsis a zero vector.
  • Rank of matrix Solving System Linear Equationsdenoted bySolving System Linear Equationsis a scalar number to determine whether the linear system is consistent (has a solution), has many solutions or has a unique set of solutions, or inconsistent (has no solution using matrix inverse). Diagram below shows the solution of the system of linear equations based on rank of the coefficient matrix Solving System Linear Equationsin comparison with the matrix size and rank of the augmented matrix coefficients Solving System Linear Equationsand the vector constants Solving System Linear Equations,Solving System Linear Equations.
  • Solving System Linear Equationshas infinitely many non-trivia solutions if and only if the matrix coefficient Solving System Linear Equationsis singular (i.e. It has no inverse, or Solving System Linear Equations), which happens when the number of equations is less than the unknowns (Solving System Linear Equations). Otherwise, homogeneous system only has unique trivia solution ofSolving System Linear Equations. General solution for homogeneous system isSolving System Linear Equationswhere Solving System Linear Equationsis an arbitrary non-zero vector.
  • The linear system Solving System Linear Equationsis called consistent ifSolving System Linear Equations. Consistent system can be solved either using matrix inverseSolving System Linear Equations, left inverseSolving System Linear Equations or right inverseSolving System Linear Equations. Full rank non-homogeneous system (happen when Solving System Linear Equations) has three possible options:
    • When the number of the unknowns in a linear system is the same as the number of equations (Solving System Linear Equations), the system is called uniquely determined system. There is only one possible solution to the system computed using matrix inverseSolving System Linear Equations.
    • When we have more equations than the unknown (Solving System Linear Equations), the system is called overdetermined system. The system is usually inconsistent with no possible solution. It is still possible to find unique solution using left inverseSolving System Linear Equations.
    • When you have more unknowns than the equations (Solving System Linear Equations), your system is called underdetermined system. The system usually has infinitely many possible solutions. The standard solution can be computed using right inverseSolving System Linear Equations.
  • When non-homogeneous system Solving System Linear Equationsis not full rank or when the rank of the matrix coefficients is less than the rank of the augmented matrix coefficients and the vector constants Solving System Linear Equationsthen the system is usually inconsistent with no possible solution using matrix inverse. It is still possible to find the approximate least square solution that minimizes the norm of errorSolving System Linear Equations using generalized inverseSolving System Linear Equations where Solving System Linear Equationsis an arbitrary non-zero vector.
Solving System Linear Equations

See also: Generalized Inverse, matrix rank, determinant, Solving Linear equations using MS Excel

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This tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\

 

 
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