# How can I prove $\hat\beta_0$ and $\hat\beta_1$ are linear in $\hat Y_i$?

A fitted regression line of a linear model is given by : $$\hat Y = \hat\beta_0 + \hat\beta_1X$$

How can I prove $$\hat\beta_0$$ and $$\hat\beta_1$$ are linear in $$\hat Y_i$$ ?

I'm unsure where to begin with this other than I know if $$\hat Y$$ is linear then $$\hat\beta_0$$ and $$\hat\beta_1$$ must also be linear.

If I can prove the beta parameters have constant slope does this imply they are linear parameters ?

If you are looking for the OLS solution as best fit for your model, then the formula for finding the vector $$\beta$$ is given by
$$$$\beta =(XX^T)^{-1} X^T Y$$$$ provided the square matrix $$XX^T$$ (here $$X^T$$ denotes the transpose of $$X$$) is not singular. This formula came from forcing the condition \begin{align} \frac{\partial L}{\partial \beta_0} &= -2\sum_{i=1}^N (y_i-(\beta_0 + \beta_1 x_i))=0,\\ \frac{\partial L}{\partial \beta_1} &= -2\sum_{i=1}^N x_i(y_i-(\beta_0 + \beta_1 x_i))=0. \end{align}
The Previous formula assures you that the relation between the vectors $$\beta$$ and $$Y$$ is linear.