## statistical context

Homogeneity is a measure of how similar many things are.

## Linear Algebra context

### …of linear maps

homogeneity is a property of Linear Maps to describe the ability to “factor out” scalars

### …of linear equations

A homogenous linear equation is one which the constant term on the right of the equations are \(0\).

#### homogenous system with more variables than equations has nonzero solutions

Proof: You can imagine the system as a matrix equation:

\begin{equation} Av = 0 \end{equation}

where, \(v\) is a list of input variables, and \(A\) is a coefficient matrix. Note that \(A = \mathbb{F}^{n} \to \mathbb{F}^{m}\), where \(n\) is the number of variables, and \(m\) the number of equations.

Now, the input variables \(v\) of the above expression is in the null space of \(A\). The question of “whether is there non-zero solutions” can be rephrased as given \(Av=0\), does \(v=0\)?" Otherwise known as “is \(null\ A=\{0\}\)?”: that is, “is \(A\) injective?”

Given the fact that map to smaller space is not injective, if \(m <n\), the map is not going to be injective. Therefore, we want \(m<n\), meaning we want more variables (\(n\)) than equations (\(m\)) to have non-zero solutions.

#### inhomogenous system with more equations than variables has no solutions for an arbitrary set of constants

Proof: You can imagine the system as a matrix equation:

\begin{equation} Av = C \end{equation}

where, \(v\) is a list of input variables, and \(A\) is a coefficient matrix. Note that \(A = \mathbb{F}^{n} \to \mathbb{F}^{m}\), where \(n\) is the number of variables, and \(m\) the number of equations.

Now, a valid solution of the above expression means that \(Av=C\) for all \(v\) (as they are, of course, the variables.) If we want the expression to have a solution for all choices of \(C\), we desire that the range of \(A\) to equal to its codomain—that we desire it to be surjective.

Given the fact that map to bigger space is not surjective, if \(m > n\), the map is not going to be surjective. Therefore, we want \(m>n\), meaning we want more equations (\(m\)) than variables (\(n\)) to have no solutions for arbitrary \(C\).