Why is the expectation of a random vector still a vector? In my previous post on derivation of covariance between y and random effect, for the following linear model:

Frank's anwer proved cov(y, u) = ZD as below:

Frank's proof involved E[u]. As far as I could understand, the random effect variable u in linear mixed model is a vector of 1*q, where q is the number of covariates in the model. My question is why the E[u] is not a single number?
What I do not understand:
If I have 3 random covariates in the model, then u MUST BE a vector of random variable of size 3*1. An example of u is as follows:

then what is E[u]? I think u is only a column of numbers, so E[u] should be the average of c(1,2,3), which is 2. However, frank says the expectation of a random vector is still a vector. I know he is right but I do not see why. I want to see the answer E[u] for my example so that I could really see what is E[u], including its dimension and how each number in E[u] is derived.

Consider $X \in \mathbb{R}^m$, i.e., $X = \left(\begin{array}{c} X_1 \\ \vdots \\ X_m \end{array}\right)$. Then, we can see that
\begin{align*}
    \mathbb{E}(X) & = \int_{\mathbb{R}^{m}} X f_{X}(X) d X\\
    & = \int_{\mathbb{R}^{m}} \left(\begin{array}{c}
       x_1\\
       \vdots \\
       x_m
    \end{array}\right) f_{X}\left(x_{1}, \ldots, x_{m}\right) d x_{1} \ldots d x_{m}\\
    & = \left(\begin{array}{c}
      \int_{\mathbb{R}^{m}} x_1 f_{X}\left(x_{1}, \ldots, x_{m}\right) d x_{1} \ldots d x_{m}\\
      \vdots\\
      \int_{\mathbb{R}^{m}} x_m f_{X}\left(x_{1}, \ldots, x_{m}\right) d x_{1} \ldots d x_{m}
    \end{array}\right)\\
    & = \left(\begin{array}{c}
      \int_{\mathbb{R}} x_1 f_{X_1}\left(x_{1}\right) d x_{1}\\
      \vdots\\
      \int_{\mathbb{R}} x_m f_{X_m}\left(x_{m}\right) d x_{m}
    \end{array}\right)\\
    & = \left(\begin{array}{c}
       \mathbb{E}(X_1)\\
       \vdots\\
       \mathbb{E}(X_m)
    \end{array}\right)
    \end{align*}

Notice that $\int_{\mathbb{R}^{m}} x_1 f_{X}\left(x_{1}, \ldots, x_{m}\right) d x_{1} \ldots d x_{m} = \int_{\mathbb{R}} \dots \int_{\mathbb{R}} x_1 f_{X}\left(x_{1}, \ldots, x_{m}\right) d x_{1} \ldots d x_{m} = \int_{\mathbb{R}} x_1 \underbrace{\left(\int_{\mathbb{R}} \dots \int_{\mathbb{R}}f_{X}\left(x_{1}, \ldots, x_{m}\right) d x_{2} \ldots d x_{m}\right)}_{f_{X_1}\left(x_{1}\right)}d x_{1}$
