If I have $(X,Y)$ with joint density $f(x,y)$ and $A$ is an invertible $2\times 2$ matrix, then for the random vector $(W,V)$ defined by:
$$ \begin{pmatrix} W\\ V \\ \end{pmatrix} = A\begin{pmatrix} X \\ Y \\ \end{pmatrix} $$
the joint density $g(w,v)$ of $(W,V)$ is given by:
$$g(w,v)=f\left(A^{-1}\begin{pmatrix} w \\ v \end{pmatrix}\right) \frac1{|\det A|} $$
If $X$ and $Y$ are independent standard normal variables and we define $Z_1=X$ and $Z_2=\rho X+\sqrt{1-\rho^2}Y$ for $-1<\rho<1$, show that $(Z_1,Z_2)$ has a standard bivariate normal distribution with parameter $\rho$.
And therefore show that if $(Z_1,Z_2)$ is a std bivariate normal random vector, the correlation of $(Z_1,Z_2)$ is $\rho$.
I'm really unsure of how I would approach this question, as I've never encountered bivariate normal distribution with matrices. I think I understand bivariate normal distributions, but I'm not sure how to apply it here. Any help would be great because I'm very, very stuck!