Question

Is there any relation between the degree of the polynomial and the number of data points in polynomial curve fitting? If yes, write the relation. If no, write a brief (no more than two lines) explanation.

Answer 1

In general, whenever you have $n$ data points, you can always find a polynomial of degree $n-1$ which passes through all of them: 

$$\begin{bmatrix} 1 & x_1 &x_1^2 & ... & x_1^{n-1} \\ 1 & x_2 &x_2^2 & ... & x_2^{n-1} \\ 1 & x_3 &x_3^2 & ... & x_3^{n-1} \\ \vdots & \vdots & \vdots & ... & \vdots \\ 1 & x_n &x_n^2 & ... & x_n^{n-1} \\ \end{bmatrix}\begin{bmatrix} c_1 \\ c_2 \\ c_3 \\ \vdots \\ c_n \end{bmatrix}=\begin{bmatrix} y_1 \\ y_2 \\ y_3 \\ \vdots \\ y_n \end{bmatrix}$$