We know that $\begin{bmatrix} B_T \\ B_t \end{bmatrix} $ is a normal Gaussian vector and $\text{Cov}(B_T, B_t) = \min\{T,t\} = t,$ i.e,
$$ \begin{bmatrix} B_T \\ B_t \end{bmatrix} \sim \mathcal{N}(\begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} T & t \\ t & t \end{bmatrix}). $$
When deriving that $E[B_t | B_T] = \frac{t}{T}B_T$, I don't understand why
$$E[B_t | B_T] = E[B_t] + \frac{\text{Cov}(B_t, B_T)}{\text{Var}(B_T)}(B_T-E[B_T])$$
Could anyone explain this step? :)