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Full linear group, Real linear group, Orthogonal group, Affine Group, Euclidean Group

Full linear group. The Full Linear Group of dimension n is the group of all non-singular n x n matrices with complex numbers as elements and matrix multiplication as the group operation.

Real linear group. The Real Linear Group of dimension n is the group of all non-singular n x n matrices with real numbers as elements and matrix multiplication as the group operation.

Orthogonal group. The Orthogonal Group of dimension n is the group of all n x n orthogonal matrices over a field F with matrix multiplication as the group operation. It is a subgroup of the Full Linear Group.

Affine Group. The Affine Group is the group of all affine transformations of the type

Y = AX + B

where A is a non-singular n x n matrix over the field F and B is a constant vector over the same field.

Euclidean Group. The Euclidean Group is the group of all affine transformations of the type

Y = AX + B

where A is an orthogonal n x n matrix over the field F and B is a constant vector over the same field. This group corresponds to the group of all rigid motions of space. It is a subgroup of the Affine Group.