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The simple bivectors e 23, e 31 and e 12 have negative squares and span the bivectors of the three-dimensional subspace corresponding to Euclidean space, R 3. These bivectors generate ordinary rotations in R 3. The simple bivectors e 14, e 24 and e 34 have positive squares and as planes span a space dimension and the time dimension. These also ...
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In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v)
The simple rotation in the zw-plane by an angle θ has bivector e 34 θ, a simple bivector. The double rotation by α and β in the xy-plane and zw-planes has bivector e 12 α + e 34 β, the sum of two simple bivectors e 12 α and e 34 β which are parallel to the two planes of rotation and have magnitudes equal to the angles of rotation.
Rotation matrices provide a simple algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics. In 2-dimensional space, a rotation can be simply described by an angle θ of rotation, but it can be also represented by the 4 entries of a rotation matrix with 2 rows and 2 columns.
Likewise, vectors whose components are contravariant push forward under smooth mappings, so the operation assigning the space of (contravariant) vectors to a smooth manifold is a covariant functor. Secondly, in the classical approach to differential geometry, it is not bases of the tangent bundle that are the most primitive object, but rather ...