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.