To qualify as a vector space, a set \(V\) and its associated operations of addition (\(+\)) and multiplication/scaling (\(\cdot\)) must adhere to the below:
Associativity # \begin{equation} \mathbf{u}+(\mathbf{v}+\mathbf{w}) = (\mathbf{u} + \mathbf{v}) + \mathbf{w} \end{equation}
Commutivity # \begin{equation} \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \end{equation}
Identity of Addition # There exists and element \(\mathbf{0}\,\in\,V\), called the zero vector, such that \(\mathbf{v} + \mathbf{0} = \mathbf{v}\) for all \(\mathbf{v}\,\in\,V\).
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An property (of a mathematical object) is invariant if, after some operation(s) are applied, that property remains unchanged.
For instance, in a geometrical space where the concept of “length” is defined (by some metric); a physical object, say a pencil, will maintain its characteristics (length) despite a change of coordinates (e.g. polar to cartesian).
In short, vectors are invariant, but their components are not (under a transformation).
A manifold is a topological space that locally resembles Euclidean space near each point.
As defined in Physics # An N-dimensional manifold of points is one for which N independent real coordinates \((x^{1}, x^{2},…,x^{N})\) are required to specify a point completely. These N coordinates are denoted collectively by \(x^{a}\), where it is understood that \(a\,=\,1,2,…,N\).
As an example, in \(\mathbb{R}^{2}\) we have a 2-dimensional manifold of points descrived by the real coordinates \((x^{1}, x^{2})\).
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This definition is more clearly described sybollically:
\begin{equation} \forall a,b \in X(aRb\,\iff\,bRa) \end{equation}
Tao’s definition:
Given any two objects \(x\) and \(y\) of the same type, if \(x = y\), then \(y = x\).
See also:
Transitive Relation
Reflexive Relation
A relation \(R\) over a set \(X\) is transitive if for all elements x, y, z in \(X\), whenever \(R\) relates \(x\) to \(y\) and \(y\) to \(z\), then \(R\) also relates \(x\) to \(z\).
\begin{equation} \forall x,y,z \in X,if\,xRy\,and\,yRz,\,then\,xRz \end{equation}
Tao’s definition:
Given any three objects x, y, and z of the same type, if \(x = y\) and \(y = z\), then \(x = z\).
See also:
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A binary relation \(R\) over a set \(X\) is reflexive if it relates every \(x \in X\) to itself.
\begin{equation} \forall x \in X \,|\, xRx \end{equation}
An example of a reflexive relation is “is equal to” since any number within \(\mathbb{R}\) would be mapped back to itself.
Tao’s definition:
Given any object x, we have \(x = x\).
See also:
Symmetric Relation
Transitive Relation
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Concepts # Relations Cardinality Binary Relation Symmetric Relation Transitive Relation Cartesian Product Reflexive Relation Surjective Function
Real Analysis Differential Geometry Linear Algebra Topology Set Theory General Concepts # Bijective Function # Injective Function # Surjective Function # Transitive Relation # Symmetric Relation # Locality # Peano Axioms #
For any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
\(z \leq x + y\)
Concepts # Metric Space Open Balls Neighborhood Manifold Topological Space