A vector space (or linear space) is the basic object of study in the branch of mathematical called linear algebra. The operations we can make between them are the vector sum and scalar multiplication, dot product, vector product and scalar triple preduce with some natural constraints such as closure of these operations, the association of these and the combination of these operations, following, we arrive at the description of a mathematical structure called space vector.
Given a formal definition body commutative scalar K (as the body of real numbers or body of complex numbers ), which call for:
0 (zero) to the null element.
1 (a) the unit element.
A set V equipped with an internal composition law (+) (vector sum), and a law External composition (·) (a scalar product) to the body K, is a vector space if and only if:
V has the structure of commutative group , regarding the internal composition law (+ ), (vector sum).
Regarding its external composition law (·), (scalar product), is fulfilled
Vector field In mathematics a vector field is a construction of calculation vector which associates a vector each point the Euclidean space .
Vector fields are often used in physical , for example, model the speed and direction of a moving fluid throughout space, or the intensity and direction of a certain strength , as the electromagnetic force or gravitational , they change point to point.
The rigorous mathematical treatment, the vector fields defined in distinguishable varieties as sections the tangent bundle variety.
Ring Given a nonempty set A and two internal laws of composition, the ordered triple (A,,) has the structure of ring if and only if
a) is associative. Ie,: a, b, c A
b) has a neutral element in A. Ie / if
c) Every element of A is invertible in A for.
is, /
d) is commutative. Ie: a, b A
These 4 properties show that (A,) is an abelian group.
e) is associative. Ie,: a, b, c A (ab) c = a (bc)
This property shows that (A,) is a semigroup.
f) distributes twice over. Ie,,: a, b, c A
a (bc) = (ab) (ac) (bc) a = (ba) (ca)
short we can say that
(A,,) is a ring iff (A,) is an abelian group, (A,) is a semigroup and the second operation distributes over the first.
DEMONSTRATION OF ROTATION MATRIX AXLE
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