Linear algebra and geometric insight: a rigorous approach to vector spaces, matrices, and linear applications

Two and a half hints from my side:

• Linear Algebra from Klaus Jänich is an introductory text which presents a short and concise introduction to the most important concepts.

All books from Jänich written for undergraduates contain many clarifying pictures (I suppose his book Topology is the most famous of them). Some students may also appreciate his sometimes unconventional style (I was one of them).

Each chapter finishes with a section of exercises to check your understanding. A few sections are explicitely dedicated for mathematicians while other sections are written with focus for physicians.

But be aware, you probably will not find all the topics you need due to the restricted length of the book.

• Finite-Dimensional Vector Spaces from Paul Halmos is my second hint for you and complementary to Jänich's book.

@Christian Blatter entitles this book the mother of modern books about Linear Algebra in the comment section and he's absolutely right. At the end of the Preface section Halmos gave credits to John von Neumann. He designates him as one of the originators of the modern spirit and methods which inspired him to present and teach the way he did in this book.

I deeply appreciate his books written from one of the great in a clear and easy to follow style. In order to get an impression of his writings I recommend his essay How to write mathematics.

• The half hint is a supplement to Halmos book, namely his Linear Algebra Problem Book which provides completely elaborated solutions to about 160 problems. If you are interested in it you may have a look at this answer.