= Solution
Take the coordinate-selection matrix
$$
A=\begin{pmatrix}I_q\\0_{(p-q)\times q}\end{pmatrix}.
$$
Then $A^TX=X_1$, $A^T\mu=\mu_1$ and $A^T\Sigma A=\Sigma_{11}$. The permitted result on a <linear image of a multivariate normal vector> therefore gives
$$
\boxed{X_1\sim N_q(\mu_1,\Sigma_{11}).}
$$
This is the <marginal distribution> obtained by discarding the other coordinates. The conclusion holds also for a positive-semidefinite marginal covariance: in that case the Gaussian law may be degenerate and need not have a full-dimensional density. If the original covariance is positive definite, its principal block $\Sigma_{11}$ is positive definite as well.
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