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:

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:

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.