Matrix Size & Validation
What is Vector?
Vector Scalar Multiple
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
What is a matrix?
Matrix Diagonal Is Diagonal Matrix?
Matrix Basic Operation
Is Equal Matrix?
Matrix Scalar Multiple
Elementary Row Operations
Finding inverse using RREF (Gauss-Jordan)
Finding Matrix Rank using RREF
Is Singular Matrix?
Matrix Generalized Inverse
Solving System of Linear Equations
Linear combination, Span & Basis Vector
Linearly Dependent & Linearly Independent
Change of basis
Matrix Nullity & Null Space
Matrix Eigen Value & Eigen Vector
Matrix Eigen Value & Eigen Vector for Symmetric Matrix
Similarity Transformation and Matrix Diagonalization
Singular Value Decomposition
Resources on Linear Algebra
Solving System Linear Equations
Linear equation is an equation in the form 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 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.
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 .
Some important notes on linear systems are:
Preferable reference for this tutorial is
Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\