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