vector algebra uses matrix multiplication to represent linear maps, and vector addition to represent translations. Formally, in the finite-dimensional case, if the linear map is represented as a multiplication by an invertible matrix and the translation as the addition of a vector , an affine map acting on a vector can be represented as Augmented matrix Using an augmented matrix and an augmented vector, it is possible to represent both the translation and the linear map using a single matrix multiplication. The technique requires that all vectors be augmented with a "1" at the end, and all matrices be augmented with an extra row of zeros at the bottom, an extra column—the translation vector—to the right, and a "1" in the lower right corner. If is a matrix, is equivalent to the following The above-mentioned augmented matrix is called an affine transformation matrix. In the general case, when the last row vector is not restricted to be , the matrix becomes a projective transformation matrix (as it can also be used to perform projective transformations). This representation exhibits the set of all invertible affine transformations as the semidirect product of and . This is a group under the operation of composition of functions, called the affine group. Ordinary matrix-vector multiplication always maps the origin to the origin, and could therefore never represent a translation, in which the origin must necessarily be mapped to some other point. By appending the additional coordinate "1" to every vector, one essentially considers the space to be mapped as a subset of a space with an additional dimension. In that space, the original space occupies the subset in which the additional coordinate is 1. Thus the origin of the original space can be found at . A translation within the original space by means of a linear transformation of the higher-dimensional space is then possible (specifically, a shear transformation). The coordinates in the higher-dimensional space are an example of homogeneous coordinates. If the original space is Euclidean, the higher dimensional space is a real projective space. The advantage of using homogeneous coordinates is that one can combine any number of affine transformations into one by multiplying the respective matrices. This property is used extensively in computer graphics, computer vision and robotics. Example augmented matrix If the vectors are a basis of the domain's projective vector space and if are the corresponding vectors in the codomain