• Vector spaces
Systems of linear equations
More on vector spaces and linear transformations
The structure of a linear transformation
Jordan canonical form
• Vector spaces with additional structure
Forms on vector spaces
Inner product spaces
Normed vector spaces and questions of analysis
A guide to further reading
Linear Algebra for the Young Mathematician is a careful, thorough, and rigorous introduction to linear algebra. It adopts a conceptual point of view, focusing on the notions of vector spaces and linear transformations, and it takes pains to provide proofs that bring out the essential ideas of the subject. It begins at the beginning, assuming no prior knowledge of the subject, but goes quite far, and it includes many topics not usually treated in introductory linear algebra texts, such as Jordan canonical form and the spectral theorem. While it concentrates on the finite-dimensional case, it treats the infinite-dimensional case as well. The book illustrates the centrality of linear algebra by providing numerous examples of its application within mathematics. It contains a wide variety of both conceptual and computational exercises at all levels, from the relatively straightforward to the quite challenging.
Readers of this book will not only come away with the knowledge that the results of linear algebra are true, but also with a deep understanding of why they are true.