Theorem describing the vector form of sulutions to a linear system.

Theorem: The general solution to a consistent linear system $Ax = b$ can be expressed in vector form as:

$$x = p + c_1 v_1 + c_2 v_2 + \cdots + c_k v_k$$

where:

• $p$ is a particular solution to $Ax = b$
• $v_1, v_2, \ldots, v_k$ are vectors that span the solution space of the homogeneous system $Ax = 0$
• $c_1, c_2, \ldots, c_k$ are free parameters (one per free variable)
• $k = n - \text{rank}(A)$ is the number of free variables

Geometric interpretation: The solution set is an affine subspace — a translation of the null space by the particular solution $p$.

How to compute:

1. Find any particular solution $p$ to $Ax = b$
2. Solve $Ax = 0$ to find the basis vectors $v_i$
3. Combine as shown above