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There is a symmetry between a function and its inverse. Specifically, if f is an invertible function with domain X and codomain Y, then its inverse f −1 has domain Y and image X, and the inverse of f −1 is the original function f. In symbols, for functions f:X → Y and f −1:Y → X, [13]
Over a field F, a matrix is invertible if and only if its determinant is nonzero. Therefore, an alternative definition of GL(n, F) is as the group of matrices with nonzero determinant. Over a commutative ring R, more care is needed: a matrix over R is invertible if and only if its determinant is a unit in R, that is, if its determinant is ...
In linear algebra, an invertible matrix is a square matrix which has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix. Invertible matrices are the same size as their ...
Moreover, f is the composition of the canonical projection from f to the quotient set, and the bijection between the quotient set and the codomain of . The composition of two surjections is again a surjection, but if g ∘ f {\displaystyle g\circ f} is surjective, then it can only be concluded that g {\displaystyle g} is surjective (see figure).
An element y is called (simply) an inverse of x if xyx = x and y = yxy. Every regular element has at least one inverse: if x = xzx then it is easy to verify that y = zxz is an inverse of x as defined in this section. Another easy to prove fact: if y is an inverse of x then e = xy and f = yx are idempotents, that is ee = e and ff = f.
A matrix is invertible (in the sense that there is an inverse matrix whose entries are in ) if and only if its determinant is an invertible element in . [43] For =, this means that the determinant is +1 or −1.
For functions of more than one variable, the theorem states that if is a continuously differentiable function from an open subset of into , and the derivative ′ is invertible at a point a (that is, the determinant of the Jacobian matrix of f at a is non-zero), then there exist neighborhoods of in and of = such that () and : is bijective. [1]
For instance, the continuously differentiable function f is invertible near a point p ∈ R n if the Jacobian determinant at p is non-zero. This is the inverse function theorem. Furthermore, if the Jacobian determinant at p is positive, then f preserves orientation near p; if it is negative, f reverses orientation.