polynomial function $y(x,w)$ is a linear function of the coefficients $w$

Question

Although polynomial function $y(x,w)$ is a nonlinear function of $x$, it is a linear function of the coefficients $w$

I came across a phrase like this here.

It might be a simple thing but I don't understand how is it so.

Answer 1

Let $$P(x,w_1)=w_{10} + w_{11}x+w_{12}x^2+....w_{1n}x^n$$ and $$P(x,w_2)=w_{20} + w_{21}x+w_{22}x^2+....w_{2n}x^n$$

Then $$P(x,w_1+w_2)= $$

$$(w_{10}+ w_{20})+( w_{11}+w_{21})x+ (w_{12}+ w_{22}) x^2+....(w_{1n}+w_{2n}) x^n=$$

$$w_{10} + w_{11}x+w_{12}x^2+....w_{1n}x^n +$$

$$w_{20} + w_{21}x+w_{22}x^2+....w_{2n}x^n=$$

$$P(x,w_1)+P(x,w_2)$$

Similarly $$P(x, \lambda w)=\lambda w_0 +\lambda w_1x+\lambda w_2x^2+....\lambda w_nx^n=$$

$$\lambda P(x, w)$$