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!