Suppose that for some application it is mathematically convenient to represent certain objects of interest (e.g., lines or conics) as n-dimensional vectors. That such a representation exists lets us conclude that in order to specify such an object, no more than n values are necessary. That is, there are at most n degrees of freedom. But now suppose I tell you that this representation has the following special property: any two vectors that are scalar multiples of one another represent the same object (provided that the scale factor is nonzero). So here’s my question: does this allow us to conclude that there are, in fact, at most n-1 degrees of freedom? If so, why? Examples To make my question more clear, here are two instances of this scenario. Lines in the plane, which have 2 DOF, can be expressed in the form ax+by+c=0, and hence we may choose to represent them as 3-vectors (a,b,c). Since (ka)x+(kb)y+(kc)=0 for nonzero k represents the same line, (a,b,c) is equivalent to all scalar multiples (ka,kb,kc). Transformations that act on homogeneous coordinates are generally defined only up to scale. For example, 3x3 homographies have 8 DOF rather than 9 "because" they are defined only up to scale.