Science:Math Exam Resources/Courses/MATH307/December 2010/Question 05 (a)
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Question 05 (a) 

Define a sequence x_{0}, x_{1}, ... by the initial conditions x_{0} = a, x_{1} = b and x_{2} = c together with the recursion relation for n = 0, 1, 2, ... (a) Rewrite this recursion in matrix form X_{n+1} = AX_{n} for n = 0, 1, 2, ... for a sequence X_{n} of vectors, with an initial vector X_{0} and some matrix A. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

What will be the (minimal) number of entries in the vector X_{n} and what will be the size of the matrix A? 
Hint 2 

The vector X_{n} must contain x_{n+2}, x_{n+1} and x_{n}, which suggest that X_{n} has length 3, and that hence A is a 3×3 matrix. 
Hint 3 

To get to three equations we can always add trivial equations such as x_{n+2} = x_{n+2}. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. As suggested in the hint we first add two trivial equations to help us rewrite the recursion relation in matrix form: This can be written as: So define and . Plugging in n = 0 we see that the initial vector X_{0} contains the initial conditions:. 