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Parametric
Solutions

Learn to interpret parametric vector forms and understand how solution sets of linear systems describe lines, planes, and higher-dimensional objects.

Linear Algebra

What Is a Parametric Solution?

When a linear system has infinitely many solutions, we cannot list them all. Instead, we express the solution set using one or more free variables as parameters. Each solution is a function of those parameters.

Free vs Pivot Variables

Pivot Variables
Correspond to pivot columns in the reduced row echelon form. They are determined by the free variables.
Free Variables
Correspond to non-pivot columns. They can take any real value and serve as the parameters.

Why Parametric Form Matters

Geometry
Reveals the shape of the solution set: lines, planes, or hyperplanes in \( \mathbb{R}^n \).
Structure
Decomposes solutions into a particular solution plus a homogeneous part.
Applications
Used in computer graphics, optimization, and describing subspaces like null spaces.

The Parametric Vector Form

The general solution of \( A\mathbf{x} = \mathbf{b} \) can be written as:

With \( k \) free variables, the solution set is a \( k \)-dimensional flat (affine subspace) in \( \mathbb{R}^n \).

Geometric Visualization

\( x_1 \)
\( x_2 \)
\( x_3 \)
\( \mathbf{p} \) (particular)
\( \mathbf{v} \) (direction)
\( \mathbf{x} = \mathbf{p} + t\mathbf{v} \)
 O

Homogeneous vs Non-Homogeneous

\( A\mathbf{x} = \mathbf{0} \) \( A\mathbf{x} = \mathbf{b} \)
Type Homogeneous Non-homogeneous
Particular solution \( \mathbf{p} = \mathbf{0} \) \( \mathbf{p} \neq \mathbf{0} \) (must find one)
Solution set passes through The origin Shifted away from origin
Parametric form \( \mathbf{x} = t_1\mathbf{v}_1 + \cdots \) \( \mathbf{x} = \mathbf{p} + t_1\mathbf{v}_1 + \cdots \)
Geometric shape Subspace (through O) Affine subspace (translated)

How to Find Parametric Form

1
Row reduce the augmented matrix \( [A \mid \mathbf{b}] \) to reduced row echelon form (RREF).
2
Identify pivot columns and free columns. Assign a parameter (\( t, s, \dots \)) to each free variable.
3
Express pivot variables in terms of the free parameters using the RREF equations.
4
Write the solution vector \( \mathbf{x} \) and separate it into \( \mathbf{p} + t\,\mathbf{v}_1 + s\,\mathbf{v}_2 + \cdots \).

Example 1: One Free Variable

Solve the system whose RREF augmented matrix is:

Example 2: Two Free Variables

RREF of the augmented matrix:

Example 3: Homogeneous System

Find the null space of \( A \) whose RREF is:

Two Parameters: a Plane

\( x_1 \)
\( x_2 \)
\( x_3 \)
 O
\( \mathbf{p} \)
\( \mathbf{v}_1 \)
\( \mathbf{v}_2 \)
\( \mathbf{x} = \mathbf{p} + s \mathbf{v}_1 + t \mathbf{v}_2 \)

Connection to the Null Space

The direction vectors \( \mathbf{v}_1, \mathbf{v}_2, \dots \) in the parametric form are exactly the vectors that span the null space \( \text{Nul}(A) \). The full solution set is:

This means the solution set of \( A\mathbf{x} = \mathbf{b} \) is a translate (shift) of the null space.

Counting Dimensions

0 free variables
Unique solution: a single point.
1 free variable
Solution set is a line in \( \mathbb{R}^n \).
2 free variables
Solution set is a plane in \( \mathbb{R}^n \).
\( k \) free variables
A \( k \)-dimensional flat (affine subspace).

Common Mistakes to Avoid

✕ Forgetting the particular solution
For \( A\mathbf{x} = \mathbf{b} \) with \( \mathbf{b} \neq \mathbf{0} \), you must include \( \mathbf{p} \). Writing only the null space part gives solutions to the wrong system.
✕ Wrong sign on coefficients
When reading off RREF, if the equation is \( x_1 + 2x_3 = 3 \), then \( x_1 = 3 - 2t \), not \( 3 + 2t \). Watch the sign flip.
✕ Confusing pivot and free columns
Pivots have leading 1s in RREF. Non-pivot columns are free. Assigning a parameter to a pivot column produces an incorrect solution.

Summary Table

Feature Description
Parametric vector form \( \mathbf{x} = \mathbf{p} + t_1\mathbf{v}_1 + \cdots + t_k\mathbf{v}_k \)
Particular solution \( \mathbf{p} \) Any single solution of \( A\mathbf{x} = \mathbf{b} \)
Direction vectors \( \mathbf{v}_i \) Basis for \( \text{Nul}(A) \)
Number of parameters \( n - \text{rank}(A) \)
Geometry (1 param) Line
Geometry (2 params) Plane
Homogeneous case \( \mathbf{p} = \mathbf{0} \), solution set is a subspace

Question 1 of 5

The RREF of an augmented matrix has pivot columns 1 and 3 out of 4 total columns (plus the augmented column). How many free parameters appear in the parametric solution?

Question 2 of 5

Given the solution \( \mathbf{x} = \begin{pmatrix} 1\\0\\-2 \end{pmatrix} + t\begin{pmatrix} 3\\1\\-1 \end{pmatrix} \), what geometric object does this describe?

Question 3 of 5

For a homogeneous system \( A\mathbf{x} = \mathbf{0} \), the particular solution \( \mathbf{p} \) is:

Question 4 of 5

The RREF is \( \left[\begin{smallmatrix} 1 & 0 & -2 \\ 0 & 1 & 1 \end{smallmatrix}\right] \) for the homogeneous system. Which vector spans the null space?

Question 5 of 5

If a system in 5 unknowns has \( \text{rank}(A) = 3 \) and is consistent, the solution set is described by how many parameters?