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Transformation Matrix
In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then T( \mathbf x ) = A \mathbf x for some m \times n matrix A, called the transformation matrix of T. Note that A has m rows and n columns, whereas the transformation T is from \mathbb^n to \mathbb^m. There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors. Uses Matrices allow arbitrary linear transformations to be displayed in a consistent format, suitable for computation. This also allows transformations to be composed easily (by multiplying their matrices). Linear transformations are not the only ones that can be represented by matrices. Some transformations that are non-linear on an n-dimensional Euclidean space R''n'' can be represented as linear transformations on the ''n''+1-dimensional space R''n''+1. These include both aff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
