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Thus we can build an n × n rotation matrix by starting with a 2 × 2 matrix, aiming its fixed axis on S 2 (the ordinary sphere in three-dimensional space), aiming the resulting rotation on S 3, and so on up through S n−1. A point on S n can be selected using n numbers, so we again have 1 / 2 n(n − 1) numbers to describe any n × n ...
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.
For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal. For such matrices, the half-vectorization is sometimes more useful than the ...
Let P and Q be two sets, each containing N points in .We want to find the transformation from Q to P.For simplicity, we will consider the three-dimensional case (=).The sets P and Q can each be represented by N × 3 matrices with the first row containing the coordinates of the first point, the second row containing the coordinates of the second point, and so on, as shown in this matrix:
(For example, the factor of 2 used for doubling a vector does not change if the vector is in spherical vs. rectangular coordinates.) However, if each vector is transformed by a matrix then the triple product ends up being multiplied by the determinant of the transformation matrix, which could be quite arbitrary for a non-rotation.
In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear map V × V → F. If V is a vector space with dual space V ∗, then the canonical evaluation map, b(f, v) = f(v) is a bilinear map from V ∗ × V to the base field. Let V and W be vector spaces over the same base field F.
That is, denoting each complex number by the real matrix of the linear transformation on the Argand diagram (viewed as the real vector space ), affected by complex -multiplication on . Thus, an m × n {\displaystyle m\times n} matrix of complex numbers could be well represented by a 2 m × 2 n {\displaystyle 2m\times 2n} matrix of real numbers.
One way to do this is to say that two sets "have the same number of elements", if and only if all the elements of one set can be paired with the elements of the other, in such a way that each element is paired with exactly one element. Accordingly, one can define two sets to "have the same number of elements"—if there is a bijection between them.