Linear Subspace

Let S denote a set of D-vectors, S =1, u2,..., uM}, where u, (i = 1,2, M) are D-vectors and M is the number of the vectors in S. A span of S is a set of points that is represented by a linear combination of the vectors included in S as follows:

A span of a basis is the whole space. Let a subset of a basis be denoted by Ssub. Then span(Ssub) is a linear subspace, and its dimension is determined by the number of vectors included in the subset, Ssub.

Affine Transformation

A D-dimensional affine transformation is a combination of a linear mapping and a translation and transfers a target D-vector, x, to a D-vector,y as follows:

where A is a D x D matrix and t is a D-vector. The former matrix, A, represents the linear mapping of a target and the latter vector, t, represents the translation.

A linear mapping,y = O[x], satisfies the following conditions:

where x1 and x2 are D-vectors and a is a scalar. When the above conditions are satisfied, then the following condition is also satisfied:

Let S = {m1, u2,..., uM} and let SO = {O1], O[u2],O[uM]}. Then, linear mapping maps the linear subspace spanned, span(S), to the other linear subspace, span(SO). A mapping represented as Ax is a linear one: O[x] = Ax.

Let e, = [0 ,, 0 , 1,0 ,, 0]r, where only the i-th component is one and the other ones are equal to zero, and let u, = Ae,. Then,


Equation (2.50) shows that A = U and that the i-th column of the matrix A is identical with u, = Ae,. Linear mappings Ax preserve the shapes and sizes of targets if and only if the mapped basis, {Ae1, Ae2, ..., AeD}, is also an orthonormal one and such that the mapping consists of rotation mapping and reflection mapping. The mapping reflects targets when det(A) < 0 and rotates when det(A) > 0. The features of linear mapping can be described more clearly using a singular value decomposition of A, which will be explained below.

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