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# Fundamental Transformations

## Coordinate Transformation

Let x = [xi,x2,... ,xD]T and y = [yi,y2,... ,yo]T denote two D-vectors. Their inner product is defined as

Let a norm of x be defined as

and the angle between the directions of x and y be denoted by в. Then, xTy = ||x||||y || cos в. Two vectors, x andy, are said to be orthogonal if в = ж/2. The inner product of xTy is equal to zero if x andy are orthogonal. If ||x|| = 1, thenx 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 {ub u2,..., uDg, 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, [xi, x2,..., xD]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 = [xi,x2,... ,xD]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 {vb v2,..., vD} and let p = yivi + y2V2 + . ..yoVD. Then,

where

and y = [ yi, y2,..., yD]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, {v1, v2,..., vD}, is orthonormal. Then the corresponding matrix, V, is orthonormal and VTV = VVT = I, where I is a unit matrix, and y = VTUx = VTp. This means the y-th component of the coordinates, y,, is equal to the inner product of the i-th basis vector and the target vector.

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