Overview
A system of linear equations is a collection of linear equations involving the same set of variables. Linear algebra provides powerful methods for understanding and solving these systems.
Standard Form
a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n &= b_1 \\
a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n &= b_2 \\
&\vdots \\
a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n &= b_m
\end{aligned}$$
## Matrix Form
The system can be written as:
$$A\mathbf{x} = \mathbf{b}$$
where:
$$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{bmatrix}$$
## Augmented Matrix
$$[A \mid \mathbf{b}] = \left[\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\ a_{21} & a_{22} & \cdots & a_{2n} & b_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \end{array}\right]$$
## Geometric Interpretation
### In 2D (Two Variables)
Each equation represents a line. Solutions are intersection points.
### In 3D (Three Variables)
Each equation represents a plane. Solutions are intersection points.
## Example
$$2x + y = 5$$
$$x - y = 1$$
Matrix form:
$$\begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 1 \end{bmatrix}$$
Augmented matrix:
$$\left[\begin{array}{cc|c} 2 & 1 & 5 \\ 1 & -1 & 1 \end{array}\right]$$
## Homogeneous Systems
A system $A\mathbf{x} = \mathbf{0}$ (where $\mathbf{b} = \mathbf{0}$) is homogeneous.
### Properties
- Always has the trivial solution $\mathbf{x} = \mathbf{0}$
- May have infinitely many solutions
- Solution set is a subspace (the null space of $A$)
### When Non-Trivial Solutions Exist
A homogeneous system $A\mathbf{x} = \mathbf{0}$ has non-trivial solutions if:
- Number of variables > number of equations
- $\det(A) = 0$ (for square systems)
- $A$ is not full column rank
## Consistent vs Inconsistent
| Type | Meaning | Condition |
|------|---------|-----------|
| Consistent | At least one solution exists | $\text{rank}(A) = \text{rank}([A \mid \mathbf{b}])$ |
| Inconsistent | No solution exists | $\text{rank}(A) < \text{rank}([A \mid \mathbf{b}])$ |
## Examples
### Example 1: Consistent System
$$x + y = 3$$
$$x - y = 1$$
Solution: $x = 2$, $y = 1$ (intersection point)
### Example 2: Inconsistent System
$$x + y = 3$$
$$x + y = 5$$
No solution (parallel lines)
### Example 3: Infinitely Many Solutions
$$x + y = 3$$
$$2x + 2y = 6$$
Solution: $x = t$, $y = 3 - t$ for any $t$ (same line)
## Solution Representation
For consistent systems, the general solution is:
$$\mathbf{x} = \mathbf{x}_p + \mathbf{x}_h$$
where:
- $\mathbf{x}_p$ is a particular solution (any solution to $A\mathbf{x} = \mathbf{b}$)
- $\mathbf{x}_h$ is the general solution to $A\mathbf{x} = \mathbf{0}$ (homogeneous part)