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MATH 105: Finite Mathematics 2
MATH 105: Finite Mathematics 2

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... A linear map a : V → V is an isometry iff one has aa∗ = a∗ a = 1. Isometries are also called ‘orthogonal transformations’. The set of isometries is a subgroup O(V ) ⊂ GL(V ), called the orthogonal group. We will also use the group SO(V ) = O(V )∩SL(V ). A linear operator a ∈ Endk V is called symmetr ...
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Numerical multilinear algebra: From matrices to tensors

... Note that the optimization is over a product of nonnegative orthants. The result extends to more general cones. ...
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Modern Map Methods in Particle Beam Physics

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Four-vector

In the theory of relativity, a four-vector or 4-vector is a vector in Minkowski space, a four-dimensional real vector space. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations, boosts (a change by a constant velocity to another inertial reference frame), and temporal and spatial inversions. Regarded as a homogeneous space, the transformation group of Minkowski space is the Poincaré group, which adds to the Lorentz group the group of translations. The Lorentz group may be represented by 4×4 matrices.The article considers four-vectors in the context of special relativity. Although the concept of four-vectors also extends to general relativity, some of the results stated in this article require modification in general relativity.
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