Here | v \ rangle is the standard basis element of \ mathbf { R } ^ n with a one in the v ^ { th } position.
32.
Intuitively, strong incoherence of a matrix U asserts that the orthogonal projections of standard basis vectors to U has magnitudes that have high likelihood if the singular vectors were distributed randomly.
33.
Are also orthogonal unit vectors, but they are not aligned with the axes of the Cartesian coordinate system, so the basis with these vectors does not meet the definition of standard basis.
34.
They are sometimes referred to as the versors of the coordinate system, and they form a set of mutually orthogonal unit vectors, typically referred to as a standard basis in linear algebra.
35.
The orbitals are thus expressed as linear combinations of basis functions, and the basis functions are one-electron functions which may or may not be centered on the Gaussian functions from standard basis sets.
36.
From that, I can see that the only 3 elements of the standard basis of \ mathbb { C } [ x, y ] that aren't in I are 1, x and y.
37.
While spherical polar coordinates are one orthogonal coordinate system for expressing vectors and tensors using polar and azimuthal angles and radial distance, the spherical basis are constructed from the standard basis and use complex numbers.
38.
In linear algebra, the set formed by these " n " versors is typically referred to as the standard basis of the corresponding Euclidean space, and each of them is commonly called a "'standard basis vector " '.
39.
In linear algebra, the set formed by these " n " versors is typically referred to as the standard basis of the corresponding Euclidean space, and each of them is commonly called a "'standard basis vector " '.
40.
Modulo minor technicalities all Iwahori Hecke algebras of finite type are cellular w . r . t . to the involution that maps the standard basis as T _ w \ mapsto T _ { w ^ {-1 } }.
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