Given $X_j=\sum_{k=1}^n c_{j,k}Y_k$, where $c$ is a matrix of constants and $Y$ the independent standard normal rv's, we need the $E[X_j|X_k]$.
I have used the following approach. Not sure if this is the right way to go about. Since they are independent, I have used $E[X_j|X_k] = E[X_j]$ since $E[Y_j]=0$ and hence $E[X_j|X_k]=0$.
$X_j$ and $X_k$ are not indepndent.
There is a well known result which says that $E(X_j|X_k)=cX_k$ for some constant $c$. [This is a property of jointly normal random variables with mean $0$]. We can compute $c$ by mutiplying both sides by $X_k$ and taking expecation. Thus $c=\frac {EX_jX_k} {EX_k^{2}}$. Can you compute $EX_jX_k$ and $EX_k^{2}$?