Definition of solution vector of a linear system

Definition: Solution Vector

The solution vector of a linear system is the n × 1 column vector containing the values of the variables that satisfy all equations simultaneously.

For the system Ax = b, the solution vector x is:

x = [x₁]
    [x₂]
    [...]
    [xₙ]

A vector x is a solution if and only if Ax = b (matrix multiplication yields the constant vector).

Example: For the system:

2x + y - z = 8
-3x - y + 2z = -11
-2x + y + 2z = -3

The solution vector is x = [2, 3, -1]ᵀ, since:

• 2(2) + 1(3) - 1(-1) = 8 ✓
• -3(2) - 1(3) + 2(-1) = -11 ✓
• -2(2) + 1(3) + 2(-1) = -3 ✓

If the system is consistent with free variables, the solution set is expressed parametrically as a particular solution plus a linear combination of vectors scaled by free parameters.