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.
**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.