# Linear transformation of multivariate normal distribution to a higher dimension?

Suppose I have transformation defined as $$Y_{q\times 1} = C_{q\times p}X_{p\times 1}$$, where $$X \sim N_p(\mu, \Sigma)$$. If $$q > p$$ how do I compute the distribution of $$Y$$, since I think the standard result $$Y \sim N_q(C\mu, C\Sigma C^T)$$ will fail as $$C\Sigma C^T$$ will be rank deficient. Please help?

Anyway, you can consider simple example with $$p=2$$, $$q=3$$. The last row of $$Y$$ then can always be represented with first two. (If rows of $$C$$ are linearly independent.)