 Nonlinearity

Nonlinearity is failure of the linearity conditions appropriate to the objects being studied. For a function $f:V\to W$ between vector spaces over the same field, being a linear map meansfor all $u,v\in V$ and all scalars $a,b$. A function is nonlinear in this sense if even one permitted choice violates the identity. The spaces and their operations are therefore part of the claim.^[ref]

For $f:\mathbb{R}\to \mathbb{R}$ with $f(x)=x^2$, choose $u=v=1$ and $a=b=1$. ThenThe extra term in $(u+v{)}^2=u^2+2uv+v^2$ explains the failure of superposition. Doubling an input also multiplies its squared output by four, rather than two.

A straight-line graph alone does not establish linearity in linear algebra. The affine map $g(x)=2x+3$ has a straight-line graph but $g(0)=3\ne 0$, whereas every linear map sends zero to zero. Its translated change $g(x+h)-g(x)=2h$ is linear in the increment $h$. This distinction matters when a physical instrument has an offset or a model is expressed relative to a nonzero operating point.

A nonlinear function can still have a useful local linear approximation. For the same square function,The derivative at3 gives the linear increment $6h$, while the exact omitted term is $h^2$. At $h=0.01$, the actual change is0.0601 and the linear prediction is0.06. At $h=1$, the actual change is7 and the prediction is6. Local accuracy over small increments does not turn the original function into a linear map. It states that the nonlinear remainder is small on a specified scale.