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- StudyBlue
- United-kingdom
- University of Leeds
- Mathematics
- Mathematics 1
- Phil
- Math1012 - Mathematics 2

Roberto F.

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131
Vector Space

A set V, whose elements we call vectors, with two operations: addition and multiplication. They must satisfy the following axioms:

A1) ∀u,v,w∈V, u+v=v+u and (u+v)+w=u+(v+w)

A2) ∀u,v∈V and a,b∈R, a(u+v)=au+av, (a+b)v=av+bv and (ab)v=a(bv)

A3) 1v=v, ∀v∈V

A4) 0v=0(vector), ∀v∈V

F(R)

The space (set) of all functions f:R->R.

Is a vector space

Mat_{m,n}

The set of all matricies of fixed size mxn

Subspace

A vector space V is a subset U⊂V which satisfies:

S1) 0(vector)∈U

S2) ∀u,v∈U and ∀a∈R, u+v∈U and au∈U.

Any subspace is also a vector space

Any plane through the origin...

Is a subspace of R^{3}

C^{∞}(R)

Functions that can be repeatedly differentiated

P(R)

Linear Dependance

Linear Combination

Linear Indepedance

Span

Basis

Dimensions

Coplanar

Field

Smooth Functions

Tuples

Linear Maps

Linear Operator

Kernel (null-space)

Image (range)

Nullity

Rank

Rank-Nullity Theorem For Matricies

Rank-Nullity Theorem for Linear Maps

Row Rank of a Matrix

Fredholm Alternative

Ordinary Differential Equation (ODE)

General Solution

Initial Value Problem

Maximal Domain of Existence

Separable Equations

Malthus' Model of Population Growth

Implicit Solution

First Order Linear Differential Equation

Integrating factor

Bernoulli Equations

First Order Homogeneous Equation

Reducible to Homogeneous

Exact Differential Equation

Thm: existence and uniqueness of solutions

reducible second-order equations

Principle of Superposition

Cauchy Equations

Singular Point Differential Equation

Wronskian

Abel's Identity

Linear Inhomogeneous Equations

Trial Functions

Variation of parameters formula

Resonance

Elementary Matrices

Row Space

Isomorphic

Eigenvalue

Eigenvector

Eigenspace

λ is an eigenvalue of T iff...

Calculating Eigenvalues

Characteristic polynomial

Calculating Eigenvectors

Calculating Eigenspace

Geometric Multiplicity

Algebraic Multiplicity

Change of Basis Matrix

Diagonal Matrix Equation

Calculating A^{n}

Linear Recurrence

Eigenvectors of a Linear Operator Corresponding to Distinct Eigenvalues are...

A matrix is diagonalisable iff...

Jordan Block

Canonical Form

Every matrix is....

Variables of Dynamical Systems

Solving Dynamical Systems For Distinct Eigenvalues

Solving Dynamical Systems For Repeated Eigenvalues

Autonomous System

Trajectories

Phase Space

Direction Field

Critical Point (Equilibrium points) of ODEs

Stability of Critical Points

Node

Fast Eigendirection

Slow Eigendirection

Saddle

Spiral

Vector Derivatives

Point Particles

Kinematics

Postiton Vector

Velocity

Acceleration

Angular Frequency, ω

Angular Velocity and Acceleration

Dimensions of Quantities

Dynamics

Linear Momentum

Newton's 1st Law

Newton's 2nd Law

Newton's 3rd Law

Rules for Newton's 3rd Law Pairs

Validity of Newtonian Mechanics

Gravitational Forces

Hooke's Law

Kinetic Energy

Work Done

Power

Potential Energy

Transcendental Functions are...

Equilibrium Point

Stable Equilibrium Points

Unstable Equilibrium Point

Velocity Trick

Solid-Solid Friction

Static Friction

Linear (Air) Resistence

Fundamental Equation of Simple Harmonic Motion

Angular Frequency, ω

Fundimental Equation of Simple Harmonic Motion (Solved)

Fundamental Equation of Dampened Harmonic Motion

Over-dampened Oscillator

Critical Oscillator

Under-dampened Oscillator

Fundamental Equation for Forced Harmonic Motion

Measure of Dampening

Linear Momentum of the system of two particles is conserved if...

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