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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 Linear equation is an equation in the form of
We may also have several equations and several unknowns that we would like to find out. A linear system is a set of 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 Thus, a linear system can be simplified into a matrix product A solution of the linear system is an ordered collection of Example: The interactive program below will help you to solve a system of linear equations NotesSome important notes on linear systems are:
See also: Generalized Inverse, matrix rank, determinant, Solving Linear equations using MS Excel Rate this tutorial or give your comments about this tutorial Preferable reference for this tutorial is Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\ |
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are the known constant.
are the 
unknown variables. The problem in linear equation is to find the values of the unknown variables that satisfy the equation.
, 
and 
we have linear equation 
and the solution is
.
linear equations each in 
unknown. We can write a linear system as
and put the coefficients of the unknowns into matrix coefficients
. The constants on the right hand side of the equation are written into vector constants
. The linear system above can be written as
numbers that satisfies the 
linear equations, which can be written in short as a vector solution
.
where matrix 
and 
. 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 
and vector constants
. Then you click “Solve Linear System Ax=b” button and the program will produce the vector solution
. 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 
.
is called non-homogeneous system when vector 
is not a zero vector. A linear system 
is called homogeneous system because vector 
is a zero vector.
denoted by
is 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 
in comparison with the matrix size and rank of the augmented matrix coefficients 
and the vector constants 
,
.
has infinitely many non-trivia solutions if and only if the matrix coefficient 
is singular (i.e. It has no inverse, or 
), which happens when the number of equations is less than the unknowns (
). Otherwise, homogeneous system only has unique trivia solution of
. General solution for homogeneous system is
where 
is an arbitrary non-zero vector.
is called consistent if
. Consistent system can be solved either using 
, 
or 
. 
) has three possible options:
), the system is called uniquely determined system. There is only one possible solution to the system computed using matrix inverse
.
), 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 inverse
.
), your system is called underdetermined system. The system usually has infinitely many possible solutions. The standard solution can be computed using right inverse
.
is 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 
then 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 error
using 
where 
is an arbitrary non-zero vector.