Explore the foundations of linear algebra in this comprehensive lecture, the first in a series that approaches the subject as "Linear Algebraic Geometry." Delve into the affine grid plane, vectors, and coordinate systems, learning essential vector operations and how to describe vectors using different coordinate systems. Tackle the fundamental problem of linear algebra: inverting linear coordinate changes. Discover general solutions in one and two dimensions, and engage with thought-provoking questions and exercises. Ideal for first-year college students, motivated high schoolers, teachers, and math enthusiasts seeking a more geometric and natural approach to this crucial mathematical field.