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A linear map is injective on the class of sparse vectors of order $s$ exactly when its null space contains no nonzero vector with at most $2s$ nonzero entries. Differences of two sparse vectors have at most $2s$ active coordinates. Conversely, splitting the support of a vector of a $2s$-sparse null vector into two parts constructs two distinct $s$-sparse vectors with the same image. The strict null space property implies sparse injectivity by applying it to both parts of such a split.

For a real vector $x$ with support of a vector $S$, this certificate is a vector $h$ whose measurements under the adjoint operator match the active sign function values and lie strictly between minus one and one on the inactive coordinates. If $A_S$ is injective, such a certificate proves that $x$ is the unique basis pursuit minimizer. For a nonzero $v\in \ker A$, injectivity ensures $v_{S^c}\ne 0$, and $0=⟨A^{\ast }h,v⟩$ implies the fixed-sign null space condition. The strict inequality is interpreted coordinatewise if $S^c$ is empty. The condition with the sign function written here is for real variables; complex basis pursuit uses the unit phases of active coordinates instead.