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Matrices and System of Linear Equations

1. A set of \(M\) Linear Equations each having \(N\) Variables (where Both \(M\) and \(N\) are \(\geq2\)) forms a System of Linear Equations as given below
   
   \(a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = k_1\)
   \(a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = k_2\)
   \(\vdots \)
   \(a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = k_{m}\)
2. The Co-efficients, Variables, and the Constants of the System of Linear Equations given above can be represented using Matrices as given in the following
   
   \(A=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix} \hspace{.5cm} X=\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \hspace{.5cm} K=\begin{bmatrix} k_1 \\ k_2 \\ \vdots \\ k_m \end{bmatrix} \)
   
   The Matrix \(A\) is called the Co-efficient Matrix. The Matrix/Vector \(X\) is called the Variable Matrix/Vector. The Matrix/Vector \(K\) is called the Constant Matrix/Vector.
   Using Matrix \(A\),\(X\) and \(K\) the System of Linear Equations can be represented using the following Matrix Equation
   
   \(AX=K\hspace{.5cm}\Rightarrow \begin{bmatrix}a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}=\begin{bmatrix} k_1 \\ k_2 \\ \vdots \\ k_m \end{bmatrix}\)
   
   
3. Based on Number of Equations \(M\) and Number of Variables \(N\), any System of Linear Equations can be of One of the following 3 Types
   1. System of Linear Equations having Number of Equations Greater than the Number of Variables (i.e. \(M>N\)).
   2. System of Linear Equations having Number of Equations Equal to the Number of Variables (i.e. \(M=N\)).
   3. System of Linear Equations having Number of Equations Lesser than the Number of Variables (i.e. \(M < N\)).
4. Based on Value of the Constants of Linear Equations, any System of Linear Equations can be of One of the following 2 Types
   1. Homogeneous System of Linear Equations: If the Constants of the All the Linear Equations (i.e. \(k_1, k_2, ..., k_m\)) are 0 then such a System of Linear Equations is called Homogeneous System of Linear Equations. These System of Linear Equations can either have only a Single Trivial Solution (i.e. value of all Variables (\(x_1, x_2, ..., x_n\) equal to 0) , or can have an Infinite Number of Non-Trivial Solutions. These Non-Trivial Solutions are themselves represented by Linear Equations of Variables (\(x_1, x_2, ..., x_n\)).
      
      Solutions of the Homogeneous System of Linear Equations is same as the NULL Space of the Coefficient Matrix.
   2. Non-Homogeneous System of Linear Equations: If Constant of at least One of the Linear Equations (i.e. \(k_1, k_2, ..., k_m\)) is Non-Zero, then such a System of Linear Equations is called Non-Homogeneous System of Linear Equations. These System of Linear Equations can either have No Solution, a Unique Solution or Infinitely Many Solutions.
      
      Non-Homogeneous System of Linear Equations with Unique Solution or Infinitely Many Solutions are also called Consistent.
      
      Non-Homogeneous System of Linear Equations with No Solution are also called Inconsistent.
      
      Also, finding solutions of the Non-Homogeneous System of Linear Equations is same as Finding whether the Constant Vector/Matrix belongs to the same Column Space as the Coefficient Matrix.
      
      If One or More Solutions Exist (i.e. for Consistent System of Linear Equations), then the Constant Vector/Matrix belongs to the same Column Space as the Coefficient Matrix.
      
      If Solution Does Not Exist (i.e. for Non-Consistent System of Linear Equations), then the Constant Vector/Matrix Does Not belong to the same Column Space as the Coefficient Matrix.
5. The following methods are generally used to Find Solutions for System of Linear Equations
   1. Elementary Row Operations/Gaussian Elimination: Can be used to find solutions for Both Homogeneous and Non-Homogeneous System of Linear Equations. Please check System of Linear Equations / Column Space Calculator using Gaussian Elimination/Elementary Row Operations.
   2. Cramer's Rule: Can be used to find solutions for Non-Homogeneous System of Linear Equations when Number of Equations is Same as Number of Variables (i.e \(M=N\)). Please check System of Linear Equations Calculator using Crammer's Rule.
   3. Matrix Inversion: Can be used to find solutions for Non-Homogeneous System of Linear Equations when Number of Equations is Same as Number of Variables (i.e \(M=N\)). Please check System of Linear Equations Calculator using Matrix Inverse.