My experience is that almost everything works exactly the same way until you hit eigenvalues.One loses the notion of an inner product. Learn more about hiring developers or posting ads with us It only takes a minute to sign up.I'm reading a theorem about the order of a finite field: But when you are working with an arbitrary field, you often don't have a notion of distance, angles, slopes, etc. Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. This isn't so nice in prime characteristic.I would say that the theory of real/complex inner products is generalized by the theory of (In particular, a quadratic form is a map from $V \to \mathbb{F}$, so cannot on its own be used to define orthogonality, while inner producst/bilinear/sesquilinear forms are all maps from $V \times V \to \mathbb{F}$. Mathematics Stack Exchange works best with JavaScript enabled I've looked around a little, but … All of the algebra essentially only depends on the fact that you are working over a field. Discrete mathematics therefore excludes topics in … Algebra is a very general term that includes a wide range of topics. The best answers are voted up and rise to the top
@tomasz I'm unable to get how $\mathbb{F}$ is a vector space over $\mathbb{F}_{p}$ implies that its size $q$ is equal to $p^{m}$.For further questions on this topic, you can try to compute the cardinal of the number of distinct basis for $V$ and then the cardinal of $GL(V)$ the space of invertible linear maps $V \to V$. I was wondering if "Finite Mathematics" would cover the requirement of "Discrete Mathematics". 21 comments. Chapter 4 defines the algebra of polynomials over a field, the ideals in that algebra, and the prime factorization of a polynomial. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. (2 answers)
I'd rather say that which direction is to be investigated first is probably a matter of taste. A browser-based software option is used for students learning linear algebra and linear programming in a finite math course. EDIT: If HS has a sequence of Algebra>Geo>AlgebraII>Trig>Calculus what would be a good sequence for Discrete and should I be taking Linear Algebra, as well. Featured on Meta Featured on Meta
But when you are working with … Linear Algebra is the study of vector spaces and linear mappings between those spaces. Also curious what the main differences are. Moreover, I know that fields of characteristic 2 are especially problematic.Which theorems from linear algebra no longer hold when we go from an infinite field to a finite field of characteristic greater than 2?Which further theorems break down (nontrivially) when we go from characteristic greater than 2 to characteristic 2?This is a rather approximative overview of what generalizations can be explored in an early course of linear algebra.The short answer is that all that does not use the fact that $\Bbb R$ is ordered, $\Bbb C$ has a norm, or that $\Bbb C=\Bbb R[i]=\overline{\Bbb R}$ carries on identically to all fields and it can, in principle and in point of fact, be taught directly as "linear algebra", instead of "$\Bbb R$-or-$\Bbb C$ linear algebra". Ideally it shouldn't make explicit reference to linear algebra or finite fields in its setup, and should require as little background as possible (the students have taken calculus, but not necessarily any other advanced math — in particular applications to group theory are out). Start here for a quick overview of the site )@tparker As long as you are not over characteristic 2 you can obtain a symmetric bilinear form from a quadratic form.
The best answers are voted up and rise to the top It only takes a minute to sign up.My question is similar but considers a less drastic generalization. …