## The Five Thousand Foot View (working draft)

I am readying myself for the GRE math subject exam in the late Spring, and therefore reviewing all four years of undergraduate mathematics at this time.

In what follows, I will be summarizing every major area of undergraduate mathematics, as follows:

1. Geometry: Plane, Elliptic, Hyperbolic, and Affine
2. Linear Algebra
3. Vector Analysis
4. Real Analysis
5. Complex Analysis
6. Topology
7. Ordinary Differential Equations
8. Fourier Analysis, Lebesgue Integration and General Transform Theory
9. Probability Theory
10. Abstract Algebra
11. Graph Theory
12. Combinatorics and Algorithmic Complexity
13. Set Theory and Transfinite Arithmetic
14. Basic & Analytic Number Theory
15. Partial Differential Equations
16. Differential Geometry

If you are a student of any subject, then I hope you have already asked yourself at times the question “What does all of this mean? What is it for?” if you are as I am, a mathematics major, then I believe that this question is particularly important. For one can too often get lost in the forest, standing amid the numberless trees, each one a bit different from its neighbors yet all oddly familiar, somehow similar, like wondering through a waking dream.

This is neither an idle nor a superfluous question. Call it “the big picture.” Call it what you will. It is important to know the gist and the connections between different areas of ones subject, and to know each area for what it really is about.

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Linear Algebra

Let’s look at linear algebra. The matrix is the lingua franca tool of linear algebra, and so linear algebra is the study of vector spaces and their transformations using matrices. After a first course, one should know the axioms that define a vector space, the algebra of matrices, how to express the structure of vector spaces using matrices, and especially how, given the basis of a vector space, to represent any transformation of that vector space using a matrix, and how to use and calculate eigenvalues and eigenvectors. Lastly, the key theorem of linear algebra is to know the 8 distinct conditions that each alone can guarantee a matrix is inevitable. And that is it, the heart of an entire semester of undergraduate linear algebra summarized in one paragraph. Again, in a nutshell: basic linear algebra is the study of the transformation of vector spaces into one another and understanding equivalent conditions under which a matrix is invertible.

What is linear algebra used for? Everything, and almost everywhere, is the short answer. Anytime you seek a first approximation to some problem, it is likely that you will be using linear algebra. In graph theory matrices are used to represent a graph’s structure in a precise and concise manner. In abstract algebra, matrices are used in the representation of groups and other algebraic structures. And linear algebra comes up in both ordinary and partial differential equations, differential geometry, knot theory, number theory, and almost everywhere else. Learn it well, know it well if you are going to be doing research in pure or applied mathematics.

If one takes a fourth year, second semester of linear algebra, often considered “part 2” of linear algebra studies, then one will likely encounter — inner product spaces, direct sums of subspaces, primary decomposition, reduction to triangular and Jordan forms, both rational and classical forms, dual spaces, orthogonal direct sums, bilinear and quadratic forms, and real normality — among the main topics.

Here are some great free full textbooks to download for reference or study:

Linear Algebra by Jim Hefferon. 499 pages. [direct link to whole book.]

Linear Algebra. David Cherney, Tom Denton & Andrew Waldron. 410 pages. [direct link to whole book]