Visualizing the Covariance Structure I

$2$ points possible $($ungraded$)$

In this problem, we develop some intuition for PCA by visualizing the covariance structure of a random Gaussian vector X~N$(((0$@$0)),((1$&$-0\mathrm{.}5$@$-0\mathrm{.}5$&$1))).$

Consider the following image:

Caption: In the above image, we graph the probability density function of X as follows. The point $($x$,$y$)$ is colored lighter if the density value at $($x$,$y$)$ is large and darker otherwise. The level curves consists of concentric ellipses centered at the origin $(0,0)^$T$.$ The vector A points along the minor axis, the vector C points along the major axis, and the vector B points at a $1350$ angle $($going clockwise$)$ to A$.($Assume that A$,$ B$,$ and C are all unit vectors.$)$

Along which direction is the variance of X the largest?

a$/$ A

b$/$ B

c$/$ C

Along which direction is the variance of X the smallest?

a$/$ A

b$/$ B

c$/$ C