How to Convert RREF to Vector Parametric Form with Free Variables

Understanding Pivot vs Free Variables

When you reduce an augmented matrix representing a consistent system of equations to Reduced Row Echelon Form (RREF), you may find columns that do not contain a pivot (the leading 1 of a row). The variables corresponding to these columns are called free variables, while the variables with pivots are basic (or pivot) variables.

Free variables can take on any value, meaning the system has infinitely many solutions. To express this infinite set of solutions, we convert the matrix back into algebra and write it in vector parametric form.

Step-by-Step Worked Example

Consider the following 3x5 augmented matrix already reduced to RREF:

Let the variables be x₁, x₂, x₃, x₄.

Step 1: Identify Basic and Free Variables

• Pivots are in Column 1 and Column 3. Thus, x₁ and x₃ are basic variables.
• Columns 2 and 4 do not have pivots. Thus, x₂ and x₄ are free variables.

Step 2: Write Equations in terms of Free Variables

Translate the non-zero rows of the matrix back into equations:

• Row 1: x1 + 2x2 - x4 = 4 ⇒ x₁ = 4 - 2x₂ + x₄
• Row 2: x3 + 3x4 = 2 ⇒ x₃ = 2 - 3x₄

Step 3: Define Parameters for Free Variables

Let's assign parameter variables to our free variables: set x₂ = s and x₄ = t, where s and t are any real numbers.

Step 4: Assemble the Vector Solution

Write the complete solution vector x as a column vector:

    [ x1 ]   [ 4 - 2s + t ]   [ 4 ]     [ -2 ]     [  1 ]
x = [ x2 ] = [     s      ] = [ 0 ] + s [  1 ] + t [  0 ]
    [ x3 ]   [  2 - 3t    ]   [ 2 ]     [  0 ]     [ -3 ]
    [ x4 ]   [     t      ]   [ 0 ]     [  0 ]     [  1 ]

This final line is the vector parametric form of the solution. The vector [4, 0, 2, 0]T is a particular solution, and the vectors scaled by s and t represent the basis of the null space (homogeneous solution).