I don't understand linear regression.
Assume the classic linear model:
$$Y = X \beta + \epsilon,\\ \epsilon \sim \mathbb{N}(0, \sigma^2 I_n), $$ where $Y$ is a vector of length $n$, $X$ is a matrix of size $n \times p$. How it's possible that $\epsilon$ is indeed distributed with a $\sigma^2 I_n$ covariance matrix? In my understanding that would imply that there is $n$ orthogonal 1-dimensional vectors, which is a nonsense, of course. On the other hand, consider the equation $$Y = X \beta + \mathbb{1} \epsilon,\\\epsilon \sim \mathbb{N}(0, \sigma^2),$$ where $\mathbb{1}$ is a vector of ones of length $n$. This would imply in turn a $\sigma^2 \mathbb{1} \mathbb{1}^T$ covariance matrix, which makes sense to me as it has only 1 non-zero eigenvalue.
What I am missing here?