Question about counting degrees of freedom

In summary, if a representation of an object exists in n-dimensional vectors, then no more than n values are necessary to specify the object. However, if the representation has the special property that scalar multiples of a vector represent the same object, then there may be fewer degrees of freedom. This is not always the case, as demonstrated by the example of points on a 1D line. Additionally, it is often more convenient to use more values than necessary to avoid special cases. In the simpler case of solving for x in \mathbb{R}^9, if all constraints are zero, only 8 constraints are needed to solve for x.
  • #1
samh
46
0
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 let's 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.
  1. 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).
  2. 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.
 
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  • #2
this kind of reminds me of a holographic universe where you have 3 dimensions but can map everything to 2.
 
  • #3
does this allow us to conclude that there are, in fact, at most n-1 degrees of freedom? If so, why?

I don't think it does. If it did, then points on a 1D line would have 0 degrees of freedom, which is clearly a contradiction, as there are infinitely many primes that you can place on the line.

Of course, I'm not even an armchair mathematician.
 
  • #4
You could, for example, divide the entire vector by, say, the first component and so represent any vector by <1, ...> with n-1 arbitrary numbers. To answer justsomeguys objection, if n were 1, every point would be represented by a single number but, since multiplying any such by a number, they all represent the same vector. The space containing a single vector does, in fact, have dimension 1. I have no idea what "primes" have to do with this.
 
  • #5
HallsofIvy said:
You could, for example, divide the entire vector by, say, the first component and so represent any vector by <1, ...> with n-1 arbitrary numbers.

... unless the first component was zero!

It is often more convenient to use more "values" than are strictly necessary to describe the degrees of freedom of the system, because it avoids a lot of special cases. For example you could write the equation of the line as y = mx + c with only two constants, but then you can't deal with vertical lines unless you add the (large) complication that m can equal "infinity".
 
  • #6
AlephZero said:
... unless the first component was zero!
This makes sense. Then it makes me wonder, since you can't assume WLOG that the first component is 1, how do we argue that there are n-1 DOF rather than n?

What about this simpler case: we want to solve for [itex]x\in\mathbb{R}^9[/itex] and to do so we collect constraints [itex]a_i^Tx=b_i[/itex]. If we can find 9 of these, we can solve for [itex]x[/itex]. But if all [itex]b_i[/itex] are zero (and we want [itex]x\neq0[/itex]) we need only collect 8 constraints! Why exactly is this? How would you prove it?
 

1. What is the concept of degrees of freedom in science?

The concept of degrees of freedom in science refers to the number of independent variables or parameters that can vary within a system. In other words, it represents the number of ways in which a system can move or change without violating any constraints or limitations.

2. How does counting degrees of freedom help in scientific research?

Counting degrees of freedom can help in scientific research by providing a measure of the complexity or flexibility of a system. It can also aid in determining the number of variables that need to be controlled or accounted for in an experiment, as well as identifying potential sources of error.

3. What is the difference between degrees of freedom in statistics and physics?

While the concept of degrees of freedom is used in both statistics and physics, there are some key differences between the two. In statistics, degrees of freedom refer to the number of independent pieces of information used to calculate a statistic, whereas in physics it represents the number of independent variables that can vary within a system.

4. How do you calculate degrees of freedom in a system?

The calculation of degrees of freedom in a system depends on the specific context and variables involved. In statistics, it is typically determined by subtracting the number of constraints or known values from the total sample size. In physics, it can involve considering the number of dimensions, constraints, and degrees of freedom for each component of the system.

5. Can degrees of freedom ever be negative?

No, degrees of freedom cannot be negative. It is a concept that represents the number of ways in which a system can vary, and therefore it must always be a positive integer. If the calculation of degrees of freedom results in a negative number, it suggests that there may be an error in the analysis or assumptions made about the system.

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