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A coordinate frame is defined by a set of basis vectors, for example, unit vectors along the three coordinate axes. This means that a rotation matrix, as a coordinate transformation, can also be viewed as defining a change of basis from one frame to another. The matrix representation of a general linear transformation is transformed from one frame to another using a so-called similarity transformation. For example, if A is the matrix representation of a given linear transformation in o₀x₀y₀z₀ and B is the representation of the same linear transformation in o₁x₁y₁z₁ then A and B are related as

(2.12)

where is the coordinate transformation between frames o₁x₁y₁z₁ and o₀x₀y₀z₀. In particular, if A itself is a rotation, then so is B, and thus the use of similarity transformations allows us to express the same rotation easily with respect to different frames.

Example 2.4.

Henceforth, whenever convenient we use the shorthand notation c_θ = cos θ, s_θ = sin θ for trigonometric functions. Suppose frames o₀x₀y₀z₀ and o₁x₁y₁z₁ are related by the rotation

If A = R_{z, θ} relative to the frame o₀x₀y₀z₀, then, relative to frame o₁x₁y₁z₁ we have

In other words, B is a rotation about the z₀-axis but expressed relative to the frame o₁x₁y₁z₁. This notion will be useful below and in later sections.