Let-k-be-a-field-and-M-a-finitely-generated-module

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Let and k

Let us for simplicity take m = k as an example.

Let x, y, z be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors.

Let k be an integer, and consider the integral

Let k be an integer that counts the steps of the algorithm, starting with zero.

Let k >= 1.

Let φ be a formula of degree k + 1 ; then we can write it as

Let A ( k ) be its Fourier transform at time 0:

* Let I consist of three elements i, j, and k with i ≤ j and i ≤ k ( not directed ).

Proof: Let d be the position of the leftmost ( most significant ) nonzero bit in the binary representation of s, and choose k such that the dth bit of x k is also nonzero.

Let ω be a generator of W ′ k .

Let k be a field ( such as the rational numbers ) and K be an algebraically closed field extension ( such as the complex numbers ), consider the polynomial ring kX n and let I be an ideal in this ring.

Let z be a primitive nth root of unity and let k be a positive integer.

Let V be a vector space over a field k. The core of the contraction operation, and the simplest case, is the natural pairing of V with its dual vector space V *.

Let n = 2 k + b, for 0 ≤ b ≤ 2 k .

Let Z be a complex submanifold of X of dimension k, and let i: Z → X be the inclusion map.

Let k be an algebraically closed field and let A n be an affine n-space over k. The polynomials ƒ in the ring k ..., x n can be viewed as k-valued functions on A n by evaluating ƒ at the points in A n , i. e. by choosing values in k for each x i .

Let k be an algebraically closed field and let P n be a projective n-space over k. Let f ∈ k ..., x n be a homogeneous polynomial of degree d. It is not well-defined to evaluate f on points in P n in homogeneous coordinates.

Let and be

Let the open enemy to it be regarded as a Pandora with her box opened ; ;

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 him be now ''!!

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 T be a linear operator on the finite-dimensional vector space V over the field F.

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 Af be the null space of Af.

Let N be a linear operator on the vector space V.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.

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 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 this be denoted by Af.

Let it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.

Let not your heart be troubled, neither let it be afraid ''.

The same God who called this world into being when He said: `` Let there be light ''!!

For those who put their trust in Him He still says every day again: `` Let there be light ''!!

Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.

Let her out, let her out -- that would be the solution, wouldn't it??

Let and field

Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function

Let S be a vector space over the real numbers, or, more generally, some ordered field.

Let R denote the field of real numbers.

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:

When the owners of the field come, they will say, ' Let us have back our field.

Let F be a field.

Let H be a Hilbert space, and let H * denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ x , defined by

Let C be the category of vector spaces K-Vect over a field K and let D be the category of algebras K-Alg over K ( assumed to be unital and associative ).

Let K be a topological field, namely a field with a topology such that addition, multiplication, and division are continuous.

Let the quantities u and v be the horizontal and vertical components of the velocity field of a steady incompressible, irrotational flow in two dimensions.

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 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 the field K be the set R of real numbers, and let the vector space V be the Euclidean space R 3 .

Let K be R, C, or any field, and let V be the set P of all polynomials with coefficients taken from the field K.

Let K be a field ( such as the field of real numbers ), and let V be a vector space over K.

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