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In the study of geometric algebras, a ** k-blade** or a

In detail:^{[1]}

- A 0-blade is a scalar.
- A 1-blade is a vector. Every vector is simple.
- A 2-blade is a
*simple*bivector. Sums of 2-blades are also bivectors, but not always simple. A 2-blade may be expressed as the wedge product of two vectors*a*and*b*: - A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors
*a*,*b*, and*c*: - In a vector space of dimension
*n*, a blade of grade*n*− 1 is called a*pseudovector*^{[2]}or an*antivector*.^{[3]} - The highest grade element in a space is called a
*pseudoscalar*, and in a space of dimension*n*is an*n*-blade.^{[4]} - In a vector space of dimension
*n*, there are*k*(*n*−*k*) + 1 dimensions of freedom in choosing a*k*-blade for 0 ≤*k*≤*n*, of which one dimension is an overall scaling multiplier.^{[5]}

A vector subspace of finite dimension *k* may be represented by the *k*-blade formed as a wedge product of all the elements of a basis for that subspace.^{[6]} Indeed, a *k*-blade is naturally equivalent to a *k*-subspace, up to a scalar factor. When the space is endowed with a volume form (an alternating *k*-multilinear scalar-valued function), such a *k*-blade may be normalized to take unit value, making the correspondence unique up to a sign.

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 pseudoscalars, in that they are elements of a one-dimensional space that is 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 of a change-of-coordinate function.

**^**Marcos A. Rodrigues (2000). "§1.2 Geometric algebra: an outline".*Invariants for pattern recognition and classification*. World Scientific. p. 3*ff*. ISBN 981-02-4278-6.**^**William E Baylis (2004). "§4.2.3 Higher-grade multivectors in Cℓ_{n}: Duals".*Lectures on Clifford (geometric) algebras and applications*. Birkhäuser. p. 100. ISBN 0-8176-3257-3.**^**Lengyel, Eric (2016).*Foundations of Game Engine Development, Volume 1: Mathematics*. Terathon Software LLC. ISBN 978-0-9858117-4-7.**^**John A. Vince (2008).*Geometric algebra for computer graphics*. Springer. p. 85. ISBN 978-1-84628-996-5.**^**For Grassmannians (including the result about dimension) a good book is: Griffiths, Phillip; Harris, Joseph (1994),*Principles of algebraic geometry*, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523. The proof of the dimensionality is actually straightforward. Take the exterior product of*k*vectors and perform elementary column operations on these (factoring the pivots out) until the top*k*×*k*block are elementary basis vectors of . The wedge product is then parametrized by the product of the pivots and the lower*k*× (*n*−*k*) block. Compare also with the dimension of a Grassmannian,*k*(*n*−*k*), in which the scalar multiplier is eliminated.**^**David Hestenes (1999).*New foundations for classical mechanics: Fundamental Theories of Physics*. Springer. p. 54. ISBN 0-7923-5302-1.

- David Hestenes; Garret Sobczyk (1987). "Chapter 1: Geometric algebra".
*Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics*. Springer. p. 1*ff*. ISBN 90-277-2561-6. - Chris Doran & Anthony Lasenby (2003).
*Geometric algebra for physicists*. Cambridge University Press. ISBN 0-521-48022-1. - 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. - R Wareham; J Cameron & J Lasenby (2005). "Applications of conformal geometric algebra to computer vision and graphics". In Hongbo Li; Peter J Olver & Gerald Sommer (eds.).
*Computer algebra and geometric algebra with applications*. Springer. p. 329*ff*. ISBN 3-540-26296-2.

- A Geometric Algebra Primer, especially for computer scientists.