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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 .
If the matrix represents a linear transformation, the column space of the matrix equals the image of this linear transformation. The column space of a matrix A is the set of all linear combinations of the columns in A. If A = [a 1 ⋯ a n], then colsp(A) = span({a 1, ..., a n}). Given a matrix A, the action of the matrix A on a vector x returns ...
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 ...
The special linear group consists of the matrices which do not change volume, while the special linear Lie algebra is the matrices which do not alter volume of infinitesimal sets. In fact, there is an internal direct sum decomposition g l n = s l n ⊕ K {\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus K} of operators/matrices ...
When : is a linear transformation between two finite-dimensional subspaces, with = and = (so can be represented by an matrix ), the rank–nullity theorem asserts that if has rank , then is the dimension of the null space of , which represents the kernel of .
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 ...
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.
Recall that M = I − P where P is the projection onto linear space spanned by columns of matrix X. By properties of a projection matrix, it has p = rank(X) eigenvalues equal to 1, and all other eigenvalues are equal to 0. Trace of a matrix is equal to the sum of its characteristic values, thus tr(P) = p, and tr(M) = n − p. Therefore,
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