Differential privacy (DP) has emerged as a powerful framework for designing algorithms that protect sensitive data. In this talk, I will present a line of work that explores the intersection of differential privacy and linear algebra, introducing efficient DP algorithms for fundamental algebraic tasks: solving systems of linear equations over arbitrary fields, linear inequalities over the reals, and computing affine spans and convex hulls. Our algorithms for solving equalities are strongly polynomial, while those for inequalities are only weakly polynomial—and this gap is provably inherent. As an application of these techniques, we obtain new efficient DP algorithms for learning halfspaces and affine subspaces, tasks central to modern machine learning. I will not assume familiarity with differential privacy and will begin by reviewing the definition in detail.