# Definition of matrix representation of a linear system

Put content here**Definition: Matrix Representation of a Linear System**
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The matrix representation of a linear system expresses the entire system compactly as:
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**`Ax = b`**
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where:
- `A` is the `m × n` coefficient matrix
- `x` is the `n × 1` solution vector (unknowns)
- `b` is the `m × 1` constant vector
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This representation is equivalent to both the original system of equations and the corresponding vector equation. The three representations are interchangeable:
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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`
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**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).

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