vector space adds a vector to every vector in the space, which means it is the transformation It is easy to show that this is a rigid transformation by showing that the distance between translated vectors equal the distance between the original vectors: A linear transformation of a vector space, , preserves linear combinations, A linear transformation can be represented by a matrix, which means where is an matrix. A linear transformation is a rigid transformation if it satisfies the condition, that is Now use the fact that the scalar product of two vectors v.w can be written as the matrix operation , where the T denotes the matrix transpose, we have Thus, the linear transformation L is rigid if its matrix satisfies the condition where is the identity matrix. Matrices that satisfy this condition are called orthogonal matrices. This condition actually requires the columns of these matrices to be orthogonal unit vectors. Matrices that satisfy this condition form a mathematical group under the operation of matrix multiplication called the orthogonal group of n×n matrices and denoted . Compute the determinant of the condition for an orthogonal matrix to obtain which shows that the matrix can have a determinant of either +1 or −1. Orthogonal matrices with determinant −1 are reflections, and those with determinant +1 are rotations. Notice that the set of orthogonal matrices can be viewed as consisting of two manifolds in separated by the set of singular matrices. The set of rotation matrices is called the special orthogonal group, and denoted . It is an example of a Lie group because it has the structure of a manifold. References Functions and mappings Kinematics Euclidean symmetries