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Concretely, in the case where the vector space has an inner product, in matrix notation these can be thought of as row vectors, which give a number when applied to column vectors. We denote this by V ∗ := Hom ( V , K ) {\displaystyle V^{*}:={\text{Hom}}(V,K)} , so that α ∈ V ∗ {\displaystyle \alpha \in V^{*}} is a linear map α : V → K ...
A vector treated as an array of numbers by writing as a row vector or column vector (whichever is used depends on convenience or context): = (), = Index notation allows indication of the elements of the array by simply writing a i, where the index i is known to run from 1 to n, because of n-dimensions. [1]
The above result can be written in index notation as follows. The elements of the matrix for an active rotation by an angle θ {\displaystyle \theta } about an axis n are given by R i j = cos θ δ i j + ( 1 − cos θ ) n i n j − sin θ ϵ i j k n k . {\displaystyle R_{ij}=\cos \theta \,\delta _{ij}+(1-\cos \theta )n_{i}n_{j ...
Note that, since x + y is a vector and α is a multi-index, the expression on the left is short for (x 1 + y 1) α 1 ⋯(x n + y n) α n. Leibniz formula For smooth functions f {\textstyle f} and g {\textstyle g} , ∂ α ( f g ) = ∑ ν ≤ α ( α ν ) ∂ ν f ∂ α − ν g . {\displaystyle \partial ^{\alpha }(fg)=\sum _{\nu \leq \alpha ...
In reverse Polish notation, [7] also known as postfix notation, all operations are entered after the operands on which the operation is performed. Reverse Polish notation is parenthesis-free, which usually leads to fewer button presses needed to perform an operation. By the use of a stack, one can enter formulas without the need to rearrange ...
Given real numbers x and y, integers m and n and the set of integers, floor and ceiling may be defined by the equations ⌊ ⌋ = {}, ⌈ ⌉ = {}. Since there is exactly one integer in a half-open interval of length one, for any real number x, there are unique integers m and n satisfying the equation
For any integer a and any positive odd integer n, the Jacobi symbol ( a / n ) is defined as the product of the Legendre symbols corresponding to the prime factors of n: ():= () (),
The notation is also used to denote the characteristic function in convex analysis, which is defined as if using the reciprocal of the standard definition of the indicator function. A related concept in statistics is that of a dummy variable .