Linear combinations, span and column space

A linear combination of vectors v⁽¹⁾, …, v⁽ⁿ⁾ multiplies each one by a scalar coefficient and adds the results: Σᵢ cᵢv⁽ⁱ⁾. The span of a set of vectors is every point you can reach this way.

The product Ax is exactly a linear combination of the columns of A, with the entries of x as the coefficients: Ax = Σᵢ xᵢ A:,ᵢ. So Ax = b has a solution precisely when b is in the span of A’s columns. That span is called the column space, or range, of A.

For Ax = b to have a solution for every b ∈ ℝᵐ, the column space must be all of ℝᵐ, so A needs at least m columns (n ≥ m). A 3 × 2 matrix, for instance, can only reach a 2-D plane inside ℝ³.