## Conditions upon overdetermined linear systems

Hi All

I have an exam tomorrow morning, I've almost completed my study guide but there are a few questions I have no idea how to answer. If someone here could give me a few pointers, or tell me how to solve it, or maybe you already know how to solve it; I can study off your solutions. Any help will be very much appreciated. Thanks all in advance.

1. The problem statement, all variables and given/known data

 11) Let V be a set of all polynomials in P2 that have a horizontal tangent at x = 0. Show that V is a subspace of P2. Find a basis for this subspace and the dimension of this subspace
 17) Let A = [1,2 ; -2,-5 ; 2,1 ; 1,-1 ; 1,-2] which is a 5x2 matrix (sorry I don't know how to code a matrix properly). What conditions must be imposed on vector b for the overdetermined linear system Ax = b (both x and b have arrows on top) to be consistent?
 18) Let B = {(-3,0,-3), (-3,2,-1), (1,6,-1)} and B' = {(-6,-6,0), (-2,-6,4), (-2,-3,7)} be bases for ℝ^3. a) Find the transition matrix from B to B' b) Compute the coordinate vector (w)b where w = (-5,8,-5) c) Use (w)b found in part (b) to compute (w)b'.
2. Relevant equations

For 18a)

u1 = k1*u'1 + k2*u'2 + k3*u'3
u2 = k1*u'1 + k2*u'2 + k3*u'3
u3 = k1*u'1 + k2*u'2 + k3*u'3
find solutions for k (constants) insert into matrix.

18c)
(w)b' by equation (w)b' = Pb→b' * (w)b

3. The attempt at a solution

For 11, I've completed this problem already but I am not confident in my answer so I attached a picture of all the work I've done. Please correct my mistakes of which I am sure I have many.

For 17, I have no idea whatsoever how to do this one? I don't even remember our professor discussing this in class. I did some googling but I can find a clear answer. Are overdetermined systems just a system with more rows than columns? I really need help on this one...

For 18, part C, I know you can find (w)b' by equation (w)b' = Pb→b' * (w)b but ofcourse I would need to know how to find parts a) and b) first. Any help here will be much appreciated.

******Update*******
I believe I have solved 11 and now 18ab and c. I have attached the solutions to this post. I still have no idea how to solve 17, someone please help...
Attached Thumbnails

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 Tags coordinate, linear alegbra, overdetermined, transition, vectors