Is the law of a random variable it's pdf and finding the law of a function of a random variable

Question

Given the multivariate normal random variable $ X \sim \mathcal{N}(\mu,\,\Sigma), \mu \in \mathbb{R}^n, \Sigma \in \mathbb{R}^{n\times n},$ we set $Y:=AX + b, A\in \mathbb{R}^{d\times n}, b \in \mathbb{R}^d.$ I am asked to find the law of $Y.$ It is unclear to me if by law of a random variable is meant the pdf, if it exists, or rather the distribution of the random variable in the sense of the push-forward measure of a random variable. Can somebody clarify this and help me by either providing some hints how to find the law of $Y$ or a solution proposal ? Assuming one would find the characteristic function of $Y.$ What should be the next step in finding the law of $Y$ ? I know how to use the characteristic function in one dimensional case, but I have doubts in using it in the multivariate case. Thanks for any help.