Search results
Results from the WOW.Com Content Network
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication.This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group.
In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to itself such that the image of every line is a line.
The linear transformation mapping x to Ax is invertible, i.e., it has an inverse under function composition. (There, again, "invertible" can equivalently be replaced with either "left-invertible" or "right-invertible".) The transpose A T is an invertible matrix. A is row-equivalent to the n-by-n identity matrix I n.
A function is bijective if and only if it is invertible; that is, a function : is bijective if and only if there is a function :, the inverse of f, such that each of the two ways for composing the two functions produces an identity function: (()) = for each in and (()) = for each in .
In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication.A linear group is a group that is isomorphic to a matrix group (that is, admitting a faithful, finite-dimensional representation over K).
The general linear group (,) consists of all invertible -by- matrices with real entries. [61] Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group.
The matrix exponential then gives us a map : (,) from the space of all n × n matrices to the general linear group of degree n, i.e. the group of all n × n invertible matrices. In fact, this map is surjective which means that every invertible matrix can be written as the exponential of some other matrix [ 9 ] (for this, it is essential to ...
This is the group of linear transformations of the vector space k n; if a basis of k n, is fixed, this is equivalent to the group of n×n invertible matrices with entries in k. It can be shown that any affine algebraic group is isomorphic to a subgroup of GL n (k). For this reason, affine algebraic groups are often called linear algebraic groups.