for computations with many variables in optimization or solving large systems in numerical linear algebra, developing efficient methods is highly desirable. this talk introduces an approach for large-scale optimization with sparse linear equality constraints that exploits computationally efficient orthogonal projections. for approximately solving large linear systems, (randomized) sketching methods are becoming increasingly popular. by recursively augmenting a deterministic sketching matrix, we develop a method with a finite termination property that compares favorably to randomized methods. moreover, we describe the construction of logical linear systems that can be used in e.g., covid-19 pooling tests, and a nonlinear least-squares method that addresses large data sizes in machine learning.