Distribution of inner product of normal random variable by a vector

Question

Suppose, that we have a random vector $\mathbf{x} \sim \mathcal{N}(\mu,\Sigma)$. What is the distribution of $(a\cdot x)$, where $a$ is a real vector?

It is known, that for a nonsingular real matrix $B$, it would be the normal distribution with

$$\mathbb{E}(\mathbf{x}) = B\mu, \text{Var}(\mathbf{x}) = B\Sigma B^T.$$

For a diagonal matrix $\Sigma$ it is also can be shown to have normal distribution. However, the general case is not that obvious.

Any tips are welcome.

Answer 1

It is a one-dimensional normal distribution.

https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Wold_theorem

This follows from an alternative definition of multi-dimensional Gaussian distribution.

EDIT: see the Wikipedia article (for future reference/anyone new to the question)

https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Definition