

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 (GaussJordan) 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 linear equations each in unknown. We can write a linear system as 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 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 Thus, a linear system can be simplified into a matrix product A solution of the linear system is an ordered collection of numbers that satisfies the linear equations, which can be written in short as a vector solution. Example: The interactive program below will help you to solve a system of 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 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 . 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\ 



