Pivots and Free Variables in Systems
Pivot columns determine core structure, while free variables generate infinite parametric solution spaces.
Leading Entries and Pivots
When a matrix is transformed into row echelon form, the first non-zero entry in each row is called a pivot or leading entry. Columns containing these pivots are pivot columns, corresponding to basic variables. Non-pivot columns correspond to free variables, which can take on any scalar value.
\begin{bmatrix} 1 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}Parameterizing Free Variables
Variables corresponding to non-pivot columns act as free parameters in the solution set. By expressing each basic pivot variable in terms of these free variables, we write the solution set in parametric vector form. This explicitly describes the geometric subspace of all solutions.
x_2 = t, \quad x_1 = 5 - 2tClassifying Solution Sets
A consistent system has a unique solution if every column of the coefficient matrix contains a pivot, leaving zero free variables. If consistent with at least one free variable, the system has infinitely many solutions. An inconsistent system occurs if a pivot lands in the augmented column.
0x_1 + 0x_2 + \dots + 0x_n = b_m \quad (b_m \neq 0)Flashcard
The rank of a matrix, equal to its number of pivot columns.
Check your understanding
A consistent linear system has 4 total variables and 2 pivot columns in its coefficient matrix. How many free variables does the system have?