An active transformation is one which actually changes the physical state of a system and makes sense even in the absence of a coordinate system whereas a passive transformation is merely a change in the coordinate system of no physical significance. The distinction between active and passive transformations is one which should always be kept in mind when working with transformations. By default, by transformation, mathematicians usually mean active transformations, while physicists could mean either.
In the case that vector quantities are considered in relation to position and displacement, as in vector fields, a similarity transformation of space is normally accompanied by a corresponding linear transformation of the other vector quantities, to preserve angles between e.g.
The transformation is linear because, as opposed to position, most vector quantities have a natural origin, e.g.
Changing the basis is a coordinate transformation, a linear transformation that can be summarized by a matrix, and is computationally the same as a mapping of points to other points keeping the bases the same: e.g.
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