To define it more clearly, for some column j that corresponds to the mapping f v j ,. A single linear map may be represented by many matrices.
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This is because the values of the elements of a matrix depend on the bases chosen. These are some examples:. It follows from this that the class of all vector spaces over a given field K , together with K -linear maps as morphisms , forms a category. The inverse of a linear map, when defined, is again a linear map.
This case is discussed in more detail below. Given again the finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the matrix multiplication , the addition of linear maps corresponds to the matrix addition , and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.
An endomorphism of V that is also an isomorphism is called an automorphism of V. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of V forms a group , the automorphism group of V which is denoted by Aut V or GL V. Since the automorphisms are precisely those endomorphisms which possess inverses under composition, Aut V is the group of units in the ring End V. The following dimension formula is known as the rank—nullity theorem :.
If V and W are finite-dimensional, bases have been chosen and f is represented by the matrix A , then the rank and nullity of f are equal to the rank and nullity of the matrix A , respectively. This is the dual notion to the kernel: just as the kernel is a sub space of the domain, the co-kernel is a quotient space of the target.
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Formally, one has the exact sequence. The dimension of the co-kernel and the dimension of the image the rank add up to the dimension of the target space. Its image consists of all sequences with first element 0, and thus its cokernel consists of the classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 it maps only the zero sequence to the zero sequence , its co-kernel has dimension 1. Its image is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1.
For a linear operator with finite-dimensional kernel and co-kernel, one may define index as:. This gives an indication of how many solutions or how many constraints one has: if mapping from a larger space to a smaller one, the map may be onto, and thus will have degrees of freedom even without constraints.
Conversely, if mapping from a smaller space to a larger one, the map cannot be onto, and thus one will have constraints even without degrees of freedom. In operator theory , the index of Fredholm operators is an object of study, with a major result being the Atiyah—Singer index theorem. No classification of linear maps could hope to be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space.
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Let V and W denote vector spaces over a field, F. Therefore, linear maps are said to be 1-co- 1-contra- variant objects, or type 1, 1 tensors. A linear transformation between topological vector spaces , for example normed spaces , may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is continuous if and only if it is bounded , for example, when the domain is finite-dimensional.
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An example of an unbounded, hence discontinuous, linear transformation is differentiation on the space of smooth functions equipped with the supremum norm a function with small values can have a derivative with large values, while the derivative of 0 is 0. A specific application of linear maps is for geometric transformations, such as those performed in computer graphics , where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrix.
Linear mappings also are used as a mechanism for describing change: for example in calculus correspond to derivatives; or in relativity, used as a device to keep track of the local transformations of reference frames. Another application of these transformations is in compiler optimizations of nested-loop code, and in parallelizing compiler techniques.
From Wikipedia, the free encyclopedia. Not to be confused with linear function. Sometimes in superficial treatments of Dirac notation, the symbol is defined alternatively as Now we turn our attention to matrix representations of operators. An operator can be characterized by its effect on the basis vectors. The action of on a basis vector yields some new vector which can be expanded in terms of the basis vectors so long as we have a complete basis set. It is easy to show that for a linear operator , the inner product for two general vectors not necessarily basis vectors and is given by Previously, we noted that , or.