Conditions upon overdetermined linear systems

[mosawi]

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.

Homework Statement

[STRIKE]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[/STRIKE]

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?

[STRIKE]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'.[/STRIKE]

Homework Equations

[STRIKE]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.[/STRIKE]

[STRIKE]18c)
(w)b' by equation (w)b' = Pb→b' * (w)b[/STRIKE]

The Attempt at a Solution

[STRIKE]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.[/STRIKE]

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...

[STRIKE]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.[/STRIKE]


******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...

 

[mighty2000]

Hi there! It's great that you're seeking help for your exam. I can offer some tips and pointers for each of the questions you've mentioned.

For question 11, it looks like you've already solved it, but just to confirm, V is indeed a subspace of P2. To show this, you could also use the subspace test, which checks if the set is closed under addition and scalar multiplication. In this case, it is, so V is a subspace. Your basis for V is correct, and the dimension of V is 2.

For question 17, yes, an overdetermined system is one with more rows than columns. In order for the system to be consistent, the vector b must be in the column space of A. This means that b can be expressed as a linear combination of the columns of A. One way to check this is to row reduce the augmented matrix [A|b] and see if it has a solution. If it does, then b is in the column space of A and the system is consistent.

For question 18, part a) can be solved by using the definition of a transition matrix, which is the matrix that transforms coordinates from one basis to another. You can set up a system of equations using the given bases and solve for the constants k1, k2, and k3. Once you have those, you can plug them into a matrix to get the transition matrix.

For part b), you can use the same method to find the coordinate vector (w)b. Plug in the coordinates of w into the transition matrix you found in part a) to get the coordinates in the B basis.

For part c), you're on the right track with the equation (w)b' = Pb→b' * (w)b. Once you have the transition matrix, you can use it to convert the coordinates of (w)b to (w)b' in the B' basis.

I hope this helps and good luck on your exam! Remember to always show your work and double check your answers. Let me know if you have any more questions or need clarification on anything. Good luck!