The proposed book would be a textbook on vector spaces aimed at the sophomore level university student. It is assumed that the audience is familiar with freshman year calculus (differentiation and integration of functions of a single variable). The book will fill a void in the current books available in this market: while maintaining a high level of mathematical rigor, the geometry of finite and infinite dimensional vector spaces will be emphasized throughout. "What does this vector space look like?" will be the main theme and will be addressed both informally (thinking of vectors as objects with magnitude and direction) and formally (through bases, coordinatization, and vector space isomorphism).
One of the shortfalls of some (though certainly not all) of the existing linear algebra texts is a blurring between vectors in Rn and points in Rn. The natural relationship which does exist between these objects will be emphasized, while still carefully distinguishing between them (especially notationally).
The standard topics in a sophomore linear algebra class will be covered. Numerous examples of the concepts will be given and, since the emphasis is on geometry, most examples will be accompanied by figures. Of course, this will require that most of the examples will be from two or three dimensional space.
One main difference in the proposed book and existing linear algebra texts is its emphasis on what I will call the "Fundamental Theorem of Vector Spaces": An n-dimensional vector space (with real scalars) is isomorphic to Rn. Most current linear algebra texts (in my opinion) don't emphasize this result nearly enough (thus my move to raise its level of importance in vector space theory by making it a "fundamental theorem"). It is this result that most deeply answers the question "What does this [n-dimensional] vector space look like?"
The attribute of the proposed book which makes it unique (to the best of my knowledge) among sophomore linear algebra texts is the inclusion of a detailed exploration of infinite dimensional vector spaces. Definitions of basis and linear combination will be modified from the finite dimensional case (to Riesz basis and series, respectively). This will allow for the extension of the Fundamental Theorem of Vector Spaces to the infinite dimensional case: Every infinite dimensional vector space (with appropriate restrictions on the existence of a basis) is isomorphic to the sequence space l2. This is the Riesz-Fisher Theorem and will require the reiteration of some calculus concepts (such as limits in a metric space) but the necessary background will be provided and all results on this topic will be given mathematically rigorous proofs. Even in this infinite dimensional setting, the idea of a vector as an object with magnitude and direction will be advocated. The geometry of the more familiar finite dimensional spaces (in terms of orthogonality and projections) will be carried over to the infinite dimensional setting. Fourier series in the space L2([-pi, pi]) will be motivated and introduced as well, though some of the details on this topic will be "beyond the scope of the text" (such as convergence in L2 norm versus pointwise convergence).
Another unique aspect of the book will be a chapter on topics which are of particular relevance in physics, astronomy and cosmology. There will be a section on Minkowski space which will give a brief axiomatic introduction to special relativity. Another section will go beyond the idea of vector spaces per se (which are all "flat"), and explore curvature and manifolds. The manifold idea will build on the vector space ideas established in earlier chapters. This will provide an opportunity to discuss (casually) general relativity. A final section will address the question "What is our universe: a vector space or a manifold?" The idea of the global topology of a manifold will be discussed and potential experiments will be mentioned which may determine this property for our universe.
As described in detail below, the proposed book will consist of eight chapters, averaging about four sections each. Each section will include exercises (both computational and theoretical) and probably an average of 2 or 3 figures.
Chapter 1. Vectors and Vector Spaces
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