**Definition: Matrix Representation of a Linear System**

The matrix representation of a linear system expresses the entire system compactly as:

**`Ax = b`**

where:
- `A` is the `m × n` coefficient matrix
- `x` is the `n × 1` solution vector (unknowns)
- `b` is the `m × 1` constant vector

This representation is equivalent to both the original system of equations and the corresponding vector equation. The three representations are interchangeable:

1. **System of equations**: `a₁₁x₁ + ... + a₁ₙxₙ = b₁`, etc.
2. **Vector equation**: `x₁a₁ + x₂a₂ + ... + xₙaₙ = b` (columns of A as vectors)
3. **Matrix equation**: `Ax = b`

**Why it matters:** The matrix representation unifies the treatment of linear systems. Properties of the matrix `A` (rank, determinant, invertibility) directly determine properties of the solution set (existence, uniqueness, dimensionality).