# Fundamental Transformations

## Coordinate Transformation

Let *x **= [xi,x _{2},... ,x_{D}]^{T}* and

*y*

*= [yi,y*denote two D-vectors. Their inner product is defined as

_{2},... ,yo]^{T}

Let a norm of *x* be defined as

and the angle between the directions of *x* and *y* be denoted by *в*. Then, *x*^{T}*y **= *||x||||y || cos *в*. Two vectors, *x* andy, are said to be orthogonal if *в = ж/2.* The inner product of *x*^{T}*y* is equal to zero if *x* and*y* are orthogonal. If ||x|| = 1, then*x* is said to be a unit vector.

A basis of a D-dimensional space is a set of *D* vectors of which linear combination can represent any D-vector in the space. Let a basis of D-dimensional space be denoted by {u_{b} *u*_{2},..., *u** _{D}g,* and let the origin of the space and a point other than the origin be denoted by

*O*and P, respectively. Let a D-vector, p, denote the location of the point, P, where

*p*

*=*

*OP*

*,*and assume that the D-vector, p, is represented by a linear combination such that

Then, *[x _{i}, x_{2},..., x*

_{D}]

^{T}is said to be the coordinates of

*x*under the employed basis. Let a

*D x D*vector that is constructed from the

*D*basis vectors be denoted by U, where

and let *x* denote a D-vector such that *x **= [x _{i},x_{2},... ,x*

_{D}]

^{T}. Then, (2.39) can be represented as follows:

An identical point in the space can be represented by different coordinates if the basis changes. Let a changed basis be denoted by {v_{b} v_{2},..., *v** _{D}}* and let

*p = yi*

*v*

*i + y*

*2*

*V*

*2*

*+ . ..yo*

*V*

*D.*Then,

where

and *y = [* yi, *y _{2},..., y_{D}]^{T}*. The change of the coordinates with respect to the change of the basis can be computed from the equation,

*p = Ux = Vy,*as follows:

It is said to be an orthonormal basis if every basis vector has a unit length and any two of the basis vectors are orthogonal. Assume that the basis, {v_{1}, *v*_{2},..., *v** _{D}*}, is orthonormal. Then the corresponding matrix,

*V*, is orthonormal and

*V*, where

^{T}V = VV^{T}= I*I*is a unit matrix, and

*y = V*This means the y-th component of the coordinates,

^{T}Ux = V^{T}p.*y,,*is equal to the inner product of the i-th basis vector and the target vector.