The easiest way to visualize a system of equations is by assuming 2 equations in \(\mathbb{R}^2\), the two dimensional plane. These equations are then graphed, and the solution to the system is where they meet. However, there is also the possibility that they never meet, or that they are the same line, and hence always meet. As more variables are added, the higher the dimension we are dealing with is, which also increases the chances of having infinitely many solutions.
There is another way of considering a system of equations, and that is mathematically, through algebra. This can be done quite easily as long as these equations are linear, or have no powers higher than or lower than 1. The definition is given below:
A Linear equation is an equation of the form \[ a_{1}x_{1}+a_{2}x_{2}+\dots+a_{n}x_{n}=b \] Here, \(a_{1}\dots a_{n}\) are real numbers called the coefficients of the equation, b is a real number called the constant term of the equation, and \(x_{1}\dots x_{n}\) are variables.
the coefficients can also be called scalars.
When many linear equations are related and need to be solved, you can put them together in a matrix instead of solving by substitution or another algebraic method. When this is done, three elementary operations can be done to the matrix to create equivalent matrices and eventually end up solving the equations. These elementary operations are:
1. Interchange the order in which the equations are listed. 2. Multiply any equation by a non-zero scalar. 3. Add a multiple of one equation to another equation.
Echelon Form: An entry of an augmented matrix is called a leading entry pr pivot entry if it is the leftmost non-zero entry of a row. An augmented matrix is in echelon form if
A column containing a pivot entry is also called a pivot column Echelon forms are not unique, and there can be different ones for the same matrix if different steps are take to get there. (the line indicates that the number to the right of the line is a constant)
Ex. \[ \begin{bmatrix} 4 & 5 & 6 & | & 5 \\ 1 & 4 & -1 & | & 2 \\ -3 & 2 & 5 & | & 1 \end{bmatrix} \] This is an augmented matrix
\[ \begin{bmatrix} 3 & 6 & 2 & | & 4 \\ 0 & -1 & 3 & | & 3 \\ 0 & 0 & 1 & | & 9 \end{bmatrix} \] This is an augmented matrix in echelon (or row echelon) form
Echelon Forms are usually found by Gaussian Elimination
This algorithm provides a method for using row operations to take a matrix to its echelon form. We begin with the matrix in its original form.
Repeat the process until there are no more non-zero rows left.
The number of pivot entries in a matrix in echelon form is called the rank of the matrix
Consider a system of \(m\) equations in \(n\) variables, and assume that the coefficient matrix has rank \(r\). Assume further that the system is consistent. 1. If \(r = n\), then the system has a unique solution. 2. If \(r < n\), then the system has infinitely many solutions, with \(n − r\) parameters.
An inconsistent system is when the there is a positive constant entry with the rest of the row filled with zeroes. This means that \(0 = n\) where \(n\) by definition is a number such that \(n\neq 0\). This is contradictory and thus the system will have no solution, which is analogous to two lines that are parallel
A matrix \(A\) in Reduced row echelon form or RREF has the following qualities:
An example of a matrix in RREF format is:
\[ A = \begin{bmatrix} 1 & 0 & 4 & 0 & | & 5 \\ 0 & 1 & -1 & 0 & | & -3 \\ 0 & 0 & 0 & 1 & | & 4 \\ 0 & 0 & 0 & 0 & | & 0 \end{bmatrix} \]