# sampling from a multivariate guassian: intuition behind using cholesky decomposition

I'm trying to understand how sampling from a multivariate gaussian works and why the cholesky decomposition is a way to do it.

Let's say we have a 25 dimensional multivariate with a 25x25 covariance matrix formed by using an RBF kernel. We want one sample which would have 25 points. One way to do this is multiply the lower triangular matrix (from a cholesky decomposition) with a 25x1 vector of sampled points from a univariate guassian.

Furthermore, the following shows the relationship when predicting f2 given f1.

This intuitively makes sense to me and I verified that the calculations using the lower triangular matrix comes out to this. However, I do not understand the intuition behind the calculations for the third point in this sample. I believe both the mean and variance of that point would be affected by f1, f2 as well as the covariances for those two points, but how exactly?

I'm not sure how to think about what the distribution for f3, f4, f5 ... fn looks like. How are those calculated and how are they affected by the distributions before them?