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

Created over 8 years ago, updated 25 days ago

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.