What is a pivot/basic variable? Free variable?
A pivot/basic variable corresponds to a pivot column. A free variable corresponds to a solution of the system for any value.
In linear algebra, a pivot variable (or basic variable) is a variable that corresponds to a pivot column in a matrix in row echelon form or reduced row echelon form. A pivot is a non-zero element chosen in a row operation that is used to zero all the other elements in the same column. In a system of linear equations, a pivot variable is a variable that corresponds to a leading coefficient in an augmented matrix. Generally, pivot variables denote values which are not 0, and other variables are expressed in terms of them.
On the other hand, a free variable is a variable in a matrix equation that can be assigned any value. Free variables occur when the number of unknowns is greater than the number of equations in a system of linear equations. In other words, a free variable is any variable that does not correspond to a pivot variable. These variables are assigned arbitrary values and the solution of the system of equations is expressed in terms of these free variables.
In a nutshell, pivot variables are those that lead to unique solutions, whereas free variables allow for infinitely many solutions.
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