Explore the fundamental concepts of vector spaces in linear algebra through 25 illustrative examples. Delve into subspaces, spans, nullspaces, linear independence, coordinates, and basis. Learn to determine whether a set forms a subspace, understand the properties of spans and nullspaces, and master the concept of linear independence. Investigate coordinate systems, including coordinate matrices and changes of coordinates. Examine the relationship between basis and dimension, and learn how to verify if a set of vectors constitutes a basis. Study the nullspace, column space, and rank of matrices, and explore infinite-dimensional vector spaces. Apply your knowledge to solve integration problems using coordinate changes in this comprehensive 3.5-hour course on vector spaces.