By algebraic operation [1] on a non-empty set C is a function that any ordered pair of two elements (or some pairs of elements [2] ) of such set is matched by exactly one item from the same set. From school mathematics we are familiar with the addition, multiplication, subtraction, integer division of natural numbers. Similarly in a set of matrices of equal dimension mxn, they all have the same number of rows and columns, we can add and in the case of an mxn matrix we can multiply by another nxp matrix. We can find the square, the cube of a square matrix.
Summary
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- 1 Definition
- 2 Properties
- 3 References
- 4 Bibliography
Definition
Given any non-empty set K , is called the algebraic operation (or law of composition ) in K , the arbitrary application σ of K x K in K. Thus, to each ordered pair (c, d) of the elements c and d of K is assigned, univocally, a third element σ (c, d) that is also in K. On more than one occasion, instead of σ (c, d) denotes c σ d. In practice, instead of σ , the binary operation in K is designated with the symbols: *, º,., +, Or even a simple juxtaposition cd. [3]
Examples
- Given two integers m and n, we define m * m = max {m, n}
- Let pyq be two rational numbers pyq pºq = m + q + pq
Partially defined
An operation * is partially defined in K, if it acts on its own part of KxK. As the case of the subtraction in the set N of the natural ones, it can be subtracted only when the minuend is not less than the subtraction; the division of the integers fits only when the dividend is a multiple of the divisor.
Properties
Consider the elements a, b, c of the non-empty set K, two operations, *, º in K.
- Closing, when for every ordered pair (a; b) there exists a single c, such that a * b = c. * Is also said to be closed in K. Such is the case of addition and multiplication in all number systems; the subtraction in Nof the naturals is not closed, neither is the division in Z of the integers. The purpose of an operation being closed has allowed the extension of the sets; from N to Z, it completely facilitates the subtraction; from Z to Q, fully facilitates division; of Q in R + the square root and the solution of an equation t 2 + p = 0, for positive rational p, demanded the expansion of R of the reals in C of the complexes, giving rise to the imaginary numbers. [4] , whom the geniuses of Newton, Descartes and others saw as oddities of fantasy [5] . And Descartes punched out the word “imaginary”.
- Commutative a * b = b * a
- Associative a * (b * c) = a * (b * c)
- Existence of a neutral element, there is an element e such that a * e = a
- Existence of symmetrical element. for each a there is another element a ‘such that a * a’ =
- Distributive, admitting that in K there are two operations * and º fits a * (bºc) = (a * b) º (aºc)
Reverse operation
In the set K, if for the operation * there is for everything its symmetric element a ‘, we will say that # is the inverse operation of * if.
a # b = a * b ‘, so we can define the subtraction in the integers and the division in the rational ones:
- so for integers ab = a + (- b), the difference minus b equals the sum of a with the opposite of b
- for rationals: a ÷ b = a × b -1, where b -1 is the multiplicative inverse of b ≠ 0; a between b is equal to a times the multiplicative inverse of b.
- the division is only distributive on the right, (a + b) ÷ c = a ÷ c + b ÷ c; this also applies to subtraction
