MATH1014 named theorems

Fermat’s Little Theorem

Let P be a prime number and a an integer not divisible by p. Then a^{p-1} = 1 (mod p)

Coset Lemma

The following are equivalent:
U+v1 = U+v2

v1 - v2 is in U

v2 is in U+v1

First Isomoprhism Theorem for linear functions:

let L be a linear map from V to W.

V/ker(f) is isomorphic to im(f)

Steinitzs Exchange Theorem

In a finite-dimensional vector space, the size of every linearly independent subset of vectors is less than or equal to the length of every finite spanning subset of vectors.

Extension to a basis Theorem

Every linearly independent subset of a finite dimensional vector space V can be extended to a basis for V.

Riesz Representation Theorem

Suppose V is a finite dimensional inner product space and L:V→F is linear. There exists some u in V unique such that L(v) = <v,u> for all v in V

Schur’s Theorem

Every linear operator on a finite dimensional vector space V over the complex numbers is upper triangulable with respect to some orthonormal basis.

Diagonisabillity Theorem

Suppose V is n dimensional and T is a linear operator on V. let a1,…,am denote the distinct eigenvalues of T. The following are equivalent:

T is diagonisable.

V has a basis consisting of only of eigenvalues of T

V = E(a1,T) + E(a2,T)+…..+E(am,T)

Spectral Theorem

Let V be an inner product space over C and T a linear operator on V. The following are equivalent:

T is normal

V has an orthonormal basis of eigenvalues of T.

T has a diagonal matrix with respect to some orthonormal basis of V