# Definition of solution vector of a linear system

Put content here**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.

# Parents

* Terminology