3. Systems of Linear Equations

 

Systems of Linear Equations

Gaussian Elimination and Row Echelon Form

Gaussian elimination is a method used to solve systems of linear equations by performing row operations on the augmented matrix of the system until it is in row echelon form.

Matrix Equations, Existence, and Uniqueness of Solutions

A system of linear equations can be represented as a matrix equation $\mathbf{Ax} = \mathbf{b}$, where $\mathbf{A}$ is the coefficient matrix, $\mathbf{x}$ is the column vector of variables, and $\mathbf{b}$ is the column vector of constants.

The system has a solution if and only if the rank of the augmented matrix $\begin{bmatrix} \mathbf{A} & \mathbf{b} \end{bmatrix}$ is equal to the rank of the coefficient matrix $\mathbf{A}$. If the system has a solution, it can be either unique or have infinitely many solutions.

Homogeneous and Non-homogeneous Systems, Consistency

A system of linear equations is homogeneous if all the constants on the right-hand side of the equations are zero. Otherwise, it is non-homogeneous.

A homogeneous system is always consistent and has at least one solution, namely the trivial solution where all variables are zero. A non-homogeneous system may be consistent "have at least one solution" or inconsistent "have no solution".