This expository talk will give an accessible introduction to the concept of persistence barcodes from topological data analysis in a way that only requires linear algebra as a prerequisite. We will start with an intuitive sketch of simplicial homology and then discuss persistent intervals in a concrete, example-driven way. We will finish by developing an efficient computational method, as well as some optimization tricks (as time permits), for persistent barcodes. The appropriate references or further readings will be provided during the talk. Note that I will be giving a similar talk, but assuming more knowledge of topology, in the Student DGT Seminar this week.