The core principle states that a vector **b** lies in the span of vectors {a₁, ..., aₙ} if and only if b can be expressed as a unique or non-unique linear combination of those basis vectors with scalar coefficients. This condition is formally equivalent to determining whether an augmented matrix representing the system [A | b] yields a consistent solution set upon Gaussian elimination without producing a pivot position in the rightmost column. The concept belongs to Linear Algebra and serves as the foundational criterion for defining vector subspaces, subspace membership tests, and the rank-nullity theorem within abstract vector spaces over fields like real numbers (ℝ) or complex numbers (ℂ).
The core principle states that a vector **b** lies in the span of vectors {a₁, ..., aₙ} if and only if b can be expressed as a unique or non-unique linear combination of those basis vectors with scal…