Solutions of a system of linear equations
Properties of eigenvalues and eigenvectors
Dimension of vector spaces, rank (R) +nullity =n
Implicit differentiation
Trig identities
Change order of a double integral
Fundamental Theorem of Calculus
Series: even, odd, zeros, differentiation at 0
Solution of linear differential equations: homogeneous and non homogeneous
Cyclic groups: G(g)={1, g, g^2, …., g^d}, degree of subgroup | d
A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, that is, a function such that for any x, y and z in M
- d(x, y) ≥ 0 (non-negativity)
- d(x, y) = 0 if and only ifx = y (identity of indiscernibles)
- d(x, y) = d(y, x) (symmetry)
- d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).
Given metric spaces (X,d1) and (Y,d2), a function ƒ:X→Y is called uniformly continuous if for every real numberε0 there exists δ0 such that for every x,y∈X with d1(x,y)δ, we have that d2(ƒ(x),ƒ(y))ε.
If X and Y are subsets of the real numbers, d1 and d2 can be the standard Euclidean norm, |·|, yielding the definition: for all ε0 there exists a δ0 such that for all x,y∈X, |x−y|δ implies |ƒ(x)−ƒ(y)|ε.
The difference between being uniformly continuous, and simply being continuous at every point, is that in uniform continuity the value of δ depends only on ε and not on the point in the domain.
It is an immediate consequence of the definitions that every uniformly continuous function is continuous.
The converse does not hold. Consider for instance the function . Let ε be any positive real number. Then uniform continuity requires the existence of a positive number δ such that for all x1,x2 with | x1 − x2 | < δ, we have | f(x1) − f(x2) | < ε. But for any positive number δ, we have f(x + δ) − f(x) = 2xδ + δ2 = (δ)(2x + δ), and for all sufficiently large x this quantity is greater than ε.
If , is uniformly continuous and is bounded, then f(S) is a bounded subset of R. In particular, the function from (0,1) to R is continuous but not uniformly continuous.
More generally, the image of a totally bounded subset under a uniformly continuous function is totally bounded. Beware that the image of a bounded subset of an arbitrary metric space under a uniformly continuous function need not be bounded. For instance, consider the identity function from the integers endowed with the discrete metric to the integers endowed with the usual Euclidean metric.
The Heine-Cantor theorem asserts that if X is compact, then every continuous f:X→Y is uniformly continuous. In particular, if a function is continuous on a closed bounded interval of the real line, it is uniformly continuous on that interval. The Darboux integrability of continuous functions follows almost immediately from the uniform continuity theorem.
If a real-valued function f is a continuous on and exists (and is finite), then f is uniformly continuous.
In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. A sequence { fn } of functionsconverges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x.
Suppose S is a set and fn: S → R are real-valued functions for every natural numbern. We say that the sequence (fn) is uniformly convergent with limit f: S → R if for every ε > 0, there exists a natural number N such that for all x in S and all n ≥ N, |fn(x) − f(x)| < ε.
Consider the sequence αn = sup|fn(x) − f(x)|. Clearly fn goes to f uniformly if and only ifαn goes to 0.
The sequence (fn) is said to be locally uniformly convergent with limit f if for every x in S, there exists an r > 0 such that (fn) converges uniformly on B(x,r) ∩ S.
Uniform convergence theorem. If is a sequence of continuous functions which converges uniformly towards the function , then is continuous as well.
For the minimal polynomial of an algebraic element of a field, see minimal polynomial (field theory).
In linear algebra, the minimal polynomial of an n-by-nmatrixA over a fieldF is the monic polynomialp(x) over F of least degree such that p(A)=0. Any other polynomial q with q(A) = 0 is a (polynomial) multiple of p.
The following three statements are equivalent:
- λ∈F is a root of p(x),
- λ is a root of the characteristic polynomial of A,
- λ is an eigenvalue of A.
The multiplicity of a root λ of p(x) is the size of the largest Jordan block corresponding to λ.
The minimal polynomial is not always the same as the characteristic polynomial. Consider the matrix 4In, which has characteristic polynomial (x − 4)n. However, the minimal polynomial is x − 4, since 4I − 4I = 0 as desired, so they are different for . That the minimal polynomial always divides the characteristic polynomial is a consequence of the Cayley–Hamilton theorem.