Sparse linear arrays obtained by thinning large uniform linear arrays are widely used in radar, wireless communications, sonar, and medical ultrasound because they enable large effective apertures and high angular resolution with fewer sensors. However, thinning also introduces array ambiguities: different sets of directions-of-arrival can produce identical measurements, making unique multisource estimation impossible. The underlying problem is closely related to determining the spark of generalized Vandermonde matrices, a fundamental open question with connections to algebraic coding theory, polynomial approximation, and control theory. This tutorial focuses on a new scalable framework for ambiguity characterization in sparse (thinned) uniform arrays. Building on fundamental results on identifiability, Kruskal rank, and ambiguity analysis, it introduces a convenient parameterization of the sparse array manifold that leads to an efficient structured low-rank matrix design problem. In contrast to earlier computationally expensive approaches, the new framework makes it possible to characterize essentially all ambiguities of large sparse arrays in a practical and systematic way. The tutorial addresses researchers from both academia and industry at different levels, including graduate students, post-docs, and senior researchers, providing both the theoretical background and efficient practical algorithms to pursue their own research in advanced sparse-array design and ambiguity analysis.