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Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.
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Let and V
Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.
Let V and W be vector spaces over the same field K. A function f: V → W is said to be a linear map if for any two vectors x and y in V and any scalar α in K, the following two conditions are satisfied:
Let us assume the bias is V and the barrier width is W. This probability, P, that an electron at z = 0 ( left edge of barrier ) can be found at z = W ( right edge of barrier ) is proportional to the wave function squared,
If V is a real vector space, then we replace V by its complexification V ⊗< sub > R </ sub > C and let g denote the induced bilinear form on V ⊗< sub > R </ sub > C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g |< sub > W </ sub > = 0.
The singles " So Far Away " and " Price to Play " came with two unreleased tracks, " Novocaine " and " Let It Out ", which were released for the special edition of the group's Chapter V, which came out in late 2005.
In early November 2005, Staind released the limited edition 2-CD / DVD set of Chapter V. The set included several rarities and fan favorites — music videos ; a complete, 36-page booklet with exclusive artwork ; an audio disc with an acoustic rendition of " This is Beetle "; the original, melodic rendition of " Reply "; the previously released B-side singles " Novocaine " and " Let It Out "; and live versions of " It's Been Awhile " and " Falling ", among many others.
Let u, v be arbitrary vectors in a vector space V over F with an inner product, where F is the field of real or complex numbers.
Let V and W be vector spaces ( or more generally modules ) and let T be a linear map from V to W. If 0 < sub > W </ sub > is the zero vector of W, then the kernel of T is the preimage of the zero subspace
Let V be a vector space over a field K, and let be a quadratic form on V. In most cases of interest the field K is either R, C or a finite field.
Let and be
Let every policeman and park guard keep his eye on John and Jane Doe, lest one piece of bread be placed undetected and one bird survive.
Let us assume that it would be possible for an enemy to create an aerosol of the causative agent of epidemic typhus ( Rickettsia prowazwki ) over City A and that a large number of cases of typhus fever resulted therefrom.
Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.
Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.
Let us take a set of circumstances in which I happen to be interested on the legislative side and in which I think every one of us might naturally make such a statement.
Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.
Let it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.
Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.
Let and finite-dimensional
Let denote the space of holomorphic sections of L. This space will be finite-dimensional ; its dimension is denoted.
Let V be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by.
* Let ρ be a unitary representation of a compact group G on a complex Hilbert space H. Then H splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of G.
Let V be a finite-dimensional vector space over a field k. The Grassmannian Gr ( r, V ) is the set of all r-dimensional linear subspaces of V. If V has dimension n, then the Grassmannian is also denoted Gr ( r, n ).
Let V be a finite-dimensional vector space over a field F and let ρ: G → GL ( V ) be a representation of a group G on V. The character of ρ is the function χ < sub > ρ </ sub >: G → F given by
Let and be the state spaces ( finite-dimensional Hilbert spaces ) of the sending and receiving ends, respectively, of a channel.
Theorem: Let V be a finite-dimensional vector space over a field F, and A a square matrix over F. Then V ( viewed as an F-module with the action of x given by A and extending by linearity ) satisfies the F-module isomorphism
Let V and W be finite-dimensional real or complex vector spaces, with n = dim W. Let A < sub > 1 </ sub >, ..., A < sub > n − 1 </ sub > be analytic functions with values in End ( V ) and b an analytic function with values in V, defined on some neighbourhood of ( 0, 0 ) in V × W. In this case, the same result holds.
Let V be a finite-dimensional real vector space and let b < sub > 1 </ sub > and b < sub > 2 </ sub > be two ordered bases for V. It is a standard result in linear algebra that there exists a unique linear transformation A: V → V that takes b < sub > 1 </ sub > to b < sub > 2 </ sub >.
Let be a compact group and let be the forgetful functor from finite-dimensional complex representations of G to complex finite-dimensional vector spaces.
Let be a category of finite-dimensional vector spaces, endowed with operations of tensor product and involution.