Definition of matrix representation of a linear system
Created almost 9 years ago, updated 5 months ago
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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:
System of equations: a₁₁x₁ + ... + a₁ₙxₙ = b₁, etc.
Vector equation: x₁a₁ + x₂a₂ + ... + xₙaₙ = b (columns of A as vectors)
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).
**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).