In the study of
geometric algebra
In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ge ...
s, a -blade or a simple -vector is a generalization of the concept of
scalars
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
and
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
s to include ''simple''
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalar (mathematics), scalars and Euclidean vector, vectors. If a scalar is considered a degree-zero quantity, and a vector is a d ...
s,
trivector
In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra of a vector space . This algebra is graded, associative and alternating, and consists of linear combinations of simple -vectors (al ...
s, etc. Specifically, a -blade is a
-vector that can be expressed as the
exterior product
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of the ...
(informally ''wedge product'') of 1-vectors, and is of ''
grade
Grade most commonly refers to:
* Grade (education), a measurement of a student's performance
* Grade, the number of the year a student has reached in a given educational stage
* Grade (slope), the steepness of a slope
Grade or grading may also ref ...
'' .
In detail:
*A 0-blade is a
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
.
*A 1-blade is a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
. Every vector is simple.
*A 2-blade is a ''simple''
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalar (mathematics), scalars and Euclidean vector, vectors. If a scalar is considered a degree-zero quantity, and a vector is a d ...
. Sums of 2-blades are also bivectors, but not always simple. A 2-blade may be expressed as the wedge product of two vectors and :
*:
*A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors , , and :
*:
*In a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
of
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
, a blade of grade is called a ''
pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...
''
or an ''
antivector An antivector is an element of grade in an ''n''-dimensional exterior algebra. An antivector is always a blade, and it gets its name from the fact that its components each involve a combination of all except one basis vector, thus being the oppos ...
''.
*The highest grade element in a space is called a ''
pseudoscalar
In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not.
Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. The ...
'', and in a space of dimension is an -blade.
*In a vector space of dimension , there are dimensions of freedom in choosing a -blade, of which one dimension is an overall scaling multiplier.
[For Grassmannians (including the result about dimension) a good book is: . The proof of the dimensionality is actually straightforward. Take vectors and wedge them together and perform elementary column operations on these (factoring the pivots out) until the top block are elementary basis vectors of . The wedge product is then parametrized by the product of the pivots and the lower block. Compare also with the dimension of a ]Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
, , in which the scalar multiplier is eliminated.
A
vector subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
of finite dimension may be represented by the -blade formed as a wedge product of all the elements of a basis for that subspace.
Indeed, a -blade is naturally equivalent to a -subspace endowed with a
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the ...
(an alternating -multilinear scalar-valued function) normalized to take unit value on the -blade.
Examples
In two-dimensional space, scalars are described as 0-blades, vectors are 1-blades, and area elements are 2-blades in this context known as
pseudoscalar
In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not.
Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. The ...
s, in that they are elements of a one-dimensional space distinct from regular scalars.
In three-dimensional space, 0-blades are again scalars and 1-blades are three-dimensional vectors, while 2-blades are oriented area elements. In this case 3-blades are called pseudoscalars and represent three-dimensional volume elements, which form a one-dimensional vector space similar to scalars. Unlike scalars, 3-blades transform according to the
Jacobian determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...
of a
change-of-coordinate function.
See also
*
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
*
Multivector
In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra of a vector space . This algebra is graded, associative and alternating, and consists of linear combinations of simple -vectors (a ...
*
Exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
*
Differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
*
Geometric algebra
In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ge ...
*
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
Notes
References
*
*
A Lasenby, J Lasenby & R Wareham(2004) ''A covariant approach to geometry using geometric algebra'' Technical Report. University of Cambridge Department of Engineering, Cambridge, UK.
*{{cite book , title=Computer algebra and geometric algebra with applications , year=2005 , page=329 ''ff'' , author=R Wareham , author2=J Cameron , author3=J Lasenby , name-list-style=amp , chapter=Applications of conformal geometric algebra to computer vision and graphics , chapter-url=https://books.google.com/books?id=uxofVAQE3LoC&pg=PA330 , editor1= Hongbo Li, editor2=Peter J Olver, editor2-link=Peter J. Olver, editor3=Gerald Sommer , isbn=3-540-26296-2 , publisher=Springer
External links
A Geometric Algebra Primer especially for computer scientists.
Geometric algebra
Vector calculus