Any set of vectors over the field of real numbers that is closed under addition and scalar multiplication defines a real vector space. What does the term "vector" mean in this context? The answer develops early in the first section, which introduces linear spaces, vectors, linear dependence, and vector products, properties and geometry. (Many readers will now see vectors from a somewhat different point of view from what an earlier experience with them might have given, particularly as they relate to geometry, transformations and invariance.) The second section is a discussion of basis vectors, beginning with their definition and proceeding with such related topics as change of basis, oblique coordinates, orthogonal bases and matrices, orthogonal transformations, and transformations relative to different bases. The third section introduces eigenvalues, eigenvectors, and the so-called characteristic equation that defines them, their geometric interpretation, symmetric transformations, the diagonalization of matrices, quadratic forms and conics. The fourth and final section of the chapter introduces tensors and their relationship to transformations. Here a distinction is made between contravariant and covariant tensors. This section concludes with a brief look at what is probably one of the most important concepts in differential geometry and general relativity the metric tensor.