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In linear algebra, linear transformations can be represented by matrices.If is a linear transformation mapping to and is a column vector with entries, then there exists an matrix , called the transformation matrix of , [1] such that: = Note that has rows and columns, whereas the transformation is from to .
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 ...
A transformation A ↦ P −1 AP is called a similarity transformation or conjugation of the matrix A. In the general linear group , similarity is therefore the same as conjugacy , and similar matrices are also called conjugate ; however, in a given subgroup H of the general linear group, the notion of conjugacy may be more restrictive than ...
As a linear transformation, every orthogonal matrix with determinant +1 is a pure rotation without reflection, i.e., the transformation preserves the orientation of the transformed structure, while every orthogonal matrix with determinant -1 reverses the orientation, i.e., is a composition of a pure reflection and a
For a matrix representing a linear map from W to V, the row operations correspond to change of bases in V and the column operations correspond to change of bases in W. Every matrix is similar to an identity matrix possibly bordered by zero rows and zero columns.
Similarity transformations which fix the origin also preserve scalar–vector multiplication and vector addition, making them linear transformations. Every origin-fixing reflection or dilation is a conformal linear transformation, as is any composition of these basic transformations, including rotations and improper rotations and most generally ...
Essentially, the matrices A and Λ represent the same linear transformation expressed in two different bases. The eigenvectors are used as the basis when representing the linear transformation as Λ. Conversely, suppose a matrix A is diagonalizable. Let P be a non-singular square matrix such that P −1 AP is some diagonal matrix D.
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...