#math/linear-algebra #math/pde #flashcards/math A function is a `homogenous function` of degree $k$ if %%::%% scaling all of its inputs by $\lambda$ scales the output by $\lambda^{k}$. $ f(\lambda x_{1}, \ldots,\lambda x_{n} )=\lambda^{k}f(x)\quad \text{for some }k\text{ not necessarily }n. $ For [[2A2 - Polynomials|polynomials]], a function is homogenous if the degree of every term sums to the same value. Consider the following homogenous polynomial with degree $k=3$ $f(x)=x^{3}+x^{2}y+xy^{2}+y^{3}$. This makes sense given we can easily pull out $\lambda^{3}$ from each term in $f(\lambda x,\lambda y)=(\lambda x)^{3}+(\lambda x)^{2}(\lambda y)+(\lambda y^{2})(\lambda x)+(\lambda y)^{3}$ to get $f(\lambda x,\lambda y)=\lambda^{3}f(x,y)$.