1) you have a 3x1 vector x of three i.i.d. Gaussian random variables.
you have a matrix A which is k x 3, where k=2, 3, 4, and in all cases the
coefficients of A are such that its rank is the minimum between k and 3.
Write the joint p.d.f. of y=Ax for al the values of k.
(Note: k=2,3 are easy but k=4 requires some thought)
2) Let x be a vector of variables, A a matrix and y a given vector. What is the solution of min_x ||Ax-y||^2
The joint distribution is the multivariate normal with mean:
and covariance matrix:
In the case k=4 the multivariate normal distribution is degenerate, see: Multivariate normal distribution - Wikipedia, the free encyclopedia