**Theorem: Existence of Solutions for Ax = b**

The matrix equation `Ax = b` has a solution if and only if `b` is a linear combination of the columns of `A`.

**Statement:** Let `A` be an `m × n` matrix with columns `a₁, a₂, ..., aₙ`. Then `Ax = b` is consistent (has a solution) if and only if there exist scalars `x₁, x₂, ..., xₙ` such that:
```
x₁a₁ + x₂a₂ + ... + xₙaₙ = b
```

**Proof sketch:** By definition of matrix-vector multiplication, `Ax = x₁a₁ + x₂a₂ + ... + xₙaₙ`. So `Ax = b` if and only if `b` can be expressed as a linear combination of the columns of `A`.

**Geometric interpretation:** The equation `Ax = b` has a solution precisely when `b` lies in the span of the columns of `A` (the column space of `A`).

**Practical test:** The system is consistent if and only if the augmented matrix `[A | b]` has no row of the form `[0 0 ... 0 | c]` where `c ≠ 0` in its echelon form.