The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A.
Created almost 9 years ago, updated 5 months ago
ID23768
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.
**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.