Linear Algebra free variables I have started taking a basic course in linear algebra and have a doubt.
I understand that in order to know if AX=B Has solutions and how many of them..first we modify the matrix to its echelon form.
I am told the number of columns free of pivots = number of free variable.
Even after giving thought to this..i cant reason this out..Any help would be much apppreciated.
Thanks in advance!
 A: Reducing a matrix to echelon form, we can see that a column is either a pivot column or a non-pivot column.
A system of linear equations corresponds to $Ax=b$. In a linear system are two types of variables: basic or free. The basic variables are determined by the pivot positions of $A$, whilst free variables are determined by the non-pivot positions of $A$.
For example, suppose we are given the equation $Ax=b$ where $A=\begin{pmatrix} 1 & 0 & 5 \\ 0 & 1 & -2 \end{pmatrix}$, $b=\begin{pmatrix} b_1 \\ b_2 \end{pmatrix}$. Notice that $A$ is in echelon form, and the equation $Ax=b$ is equivalent to:
$$Ax=\begin{pmatrix} \color{red}{1} & 0 & \color{blue}{5} \\ 0 & \color{red}{1} & \color{blue}{-2} \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}=\begin{pmatrix} b_1 \\ b_2 \end{pmatrix}$$
and is equivalent to the system of linear equations:
$$\begin{align}\color{red}{x_1}+0x_2+\color{blue}{5x_3}&=b_1\\
0x_1+\color{red}{x_2}+\color{blue}{(-2)x_3}&=b_2\end{align}$$
The correspondence of the pivot positions to basic variables are in red and of non-pivot positions to free variables are in blue.
The solution to this system is given by $x_1=-5x_3, x_2=2x_3$, and a free variable $x_3$. Since we have a free variable, this system of linear equation has infinitely many solution since we can always find an $x_3$ that will satisfy the equation, and since the free variable corresponds to the non-pivot positions of $A$, hence we can place any value there as well that will satisfy the matrix equation.
(Also remember that a system of linear equation either has no solution, one solution, or infinitely many solutions).
