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Transposition (mathematics), a permutation which exchanges two elements and keeps all others fixed; Transposition, producing the transpose of a matrix A T, which is computed by swapping columns for rows in the matrix A; Transpose of a linear map; Transposition (logic), a rule of replacement in philosophical logic
The transpose of a matrix A, denoted by A T, [3] ⊤ A, A ⊤, , [4] [5] A′, [6] A tr, t A or A t, may be constructed by any one of the following methods: Reflect A over its main diagonal (which runs from top-left to bottom-right) to obtain A T; Write the rows of A as the columns of A T; Write the columns of A as the rows of A T
In logic and mathematics, contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as § Proof by contrapositive. The contrapositive of a statement has its antecedent and consequent inverted and flipped.
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of + being , for real numbers and ).
The property of fours of matrices gives the possibility to create matrix with mutually orthogonal rows and columns (matrix ) by changing the sign to an odd number of elements in every one of fours (,,,,,), ,,, [,].
Let # denote the algebraic dual space of a vector space . Let and be vector spaces over the same field . If : is a linear map, then its algebraic adjoint or dual, [1] is the map #: # # defined by .
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}
(This holds as long as the cost of a transposition, , is at least the average of the cost of an insertion and deletion, i.e., +. [9]) Thus, we need to consider only two symmetric ways of modifying a substring more than once: (1) transpose letters and insert an arbitrary number of characters between them, or (2) delete a sequence of characters ...