Designed for a first year graduate course in Mechanics, this text brings together never before collected works on linear vector spaces, on which the author is a world renowned authority. It is primarily concerned with finite dimensional real Euclidean spaces, with Cartesian tensors viewed as linear transformations of such a space into itself, and with applications of these notions, especially in mechanics. The geometric content of the theory and the distinction between matrices and tensors are emphasized, and absolute- and component- notation are both employed. Problems and solutions are included.