I am currently reading a book "A first course in Linear model Theory". A linear regression model is defined as $$ y = X\beta+\epsilon$$ where $y = (Y_i, ... , Y_N )$ is an N-dimensional vector of observed responses,$\beta = ( \beta_0,\beta_1,...\beta_n)$'is a(k+1)-dimensional vector of unknown parameters,X is an $N*(k+1)$ matrix of rank r of known predictors, and $\epsilon= (\epsilon_1,\epsilon_2,...,\epsilon_n)$ is N-dimensional random vector of unobserved errors. Suppose,
$E(\epsilon)=0$ and $cov(\epsilon)=\delta ^2 I_N$ (1)
The book then writes :
Using (1) and the properties of the expectation and covariance operators:
$E(y)=E(X\beta+\epsilon)=X\beta+E(\epsilon)=X\beta$ (2)
$Cov(y)=Cov(X\beta+\epsilon)=Cov(\epsilon)=\delta ^2I_N$ (3)
I would appreciate so much a proof on how (1) and properties of expectation and covariance lead to equation (2) and (3) conclusions above