Here's what I would have submitted:


Question 1:

The eigenvalues are 1, 2, and 0 with corresponding eigenvectors e1,
e2, e3.  Thus a basis for the solution space is:
     [1]       [0]  [0]
{e^t [0], e^2t [1], [0]}.
     [0]       [0]  [1]



Question 2:

This is 1 * 3 + 3 * -1 = 3-3 = 0.



Question 3:

We have 

1 = (e1,v1) = (av1+bv2,v1) = a(v1,v1) + b(v1,v2) = a(v1,v1) = 10a

and 

3 = (e1,v2) = (av1+bv2,v2) = a(v1,v2) + b(v2,v2) = b(v2,v2) = 10b.

Thus a = 1/10 and b = 3/10.