# Linear Subspace

Let *S* denote a set of D-vectors, *S =* {и_{1}, *u _{2},..., u*

_{M}}, 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 S_{sub}. Then span(S_{sub}) is a linear subspace, and its dimension is determined by the number of vectors included in the subset, *S*sub.

# 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 x_{1} and *x** _{2}* are D-vectors and

*a*is a scalar. When the above conditions are satisfied, then the following condition is also satisfied:

Let *S =* {m_{1}, *u*_{2}*,..., u*_{M}} and let *S _{O} = {*

*O*[и

_{1}],

*O*[u

_{2}],

*O*

*[u*Then, linear mapping maps the linear subspace spanned, span(S), to the other linear subspace, span(S

_{M}]}._{O}). 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,

where

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, {Ae_{1}, Ae_{2}, ..., Ae_{D}}, 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.