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In a rigidity matroid for a graph with n vertices in d-dimensional space, a set of edges that defines a subgraph with k degrees of freedom has matroid rank dn − k. A set of edges is independent if and only if, for every edge in the set, removing the edge would increase the number of degrees of freedom of the remaining subgraph. [1] [2] [3]
In other words, a sequence of vectors is linearly independent if the only representation of as a linear combination of its vectors is the trivial representation in which all the scalars are zero. [2] Even more concisely, a sequence of vectors is linearly independent if and only if 0 {\displaystyle \mathbf {0} } can be represented as a linear ...
Kinematics is often described as applied geometry, where the movement of a mechanical system is described using the rigid transformations of Euclidean geometry. The coordinates of points in a plane are two-dimensional vectors in R 2 (two dimensional space).
It has two bases, which are the sets {(0,1),(2,0)} , {(0,3),(2,0)}. These are the only independent sets that are maximal under inclusion. The basis has a specialized name in several specialized kinds of matroids: [1] In a graphic matroid, where the independent sets are the forests, the bases are called the spanning forests of the graph.
One may compare linear motion to general motion. In general motion, a particle's position and velocity are described by vectors, which have a magnitude and direction. In linear motion, the directions of all the vectors describing the system are equal and constant which means the objects move along the same axis and do not change direction.
All points in the body have the same component of the velocity along the axis, however the greater the distance from the axis the greater the velocity in the plane perpendicular to this axis. Thus, the helicoidal field formed by the velocity vectors in a moving rigid body flattens out the further the points are radially from the twist axis.
The transport theorem (or transport equation, rate of change transport theorem or basic kinematic equation or Bour's formula, named after: Edmond Bour) is a vector equation that relates the time derivative of a Euclidean vector as evaluated in a non-rotating coordinate system to its time derivative in a rotating reference frame.
For example, in geometry, two linearly independent vectors span a plane. To express that a vector space V is a linear span of a subset S , one commonly uses one of the following phrases: S spans V ; S is a spanning set of V ; V is spanned or generated by S ; S is a generator set or a generating set of V .