Vectors + Matrices
1. The surface of a cylindrical water pipe is given in (rather useless!) 3-dimensional coordinates by the equation q(x1,x2,x3) = 8×2 1 −4x1x2 + 4x1x3 + 5×2 2 + 8x2x3 +
5×2 3 = 1. Your job is to find some good coordinates (y1,y2,y3).
(a) (2 marks) Write down the symmetric matrix Q corresponding to the quadratic form q(x1,x2,x3). (b) (6 marks) Find the eigenvalues of Q. Show your work. (c) (6 marks)
Find an orthonormal set of eigenvectors of Q. Show your work. (d) (2 marks) Write down V and D where Q = V DV T is an orthogonal diagonalization. (e) (2 marks) Let y =
V Tx where y = y1 y2 y3 and x = x1 x2 x3 . What is the quadratic form in the coordinates y1,y2,y3? (f) (2 marks) What is the radius of the pipe? (Hint: a
circle of radius r in (s,t) coordinates satisfies s2 + t2 = r2.)
2. Rats and possums infest the NZ native bush and compete with each other for food and water. Let rn and pn denote the rat and possum populations at the end of n
years. In the absence of possums the rat population grows to rn+1 = 3.5rn, but competition from possums reduces it to rn+1 = 3.5rn − pn. Possums are similarly affected
by rats, and their equation is given by pn+1 = 2.5pn −0.25rn. Suppose that initially there are 10 possums and 10 rats in the bush. (a) (1 marks) The populations after
n years can be modelled by the linear system xn+1 = Axn where xn =pn rn . What is A? (b) (2 marks) The matrix A has eigenvalues 3.707 and 2.292 (to 3 decimal
places). Use this to find matrices D and V that diagonalize A. Give your answers to 3 decimal places. (c) (2 marks) Compute the inverse V −1 to 3 decimal places. (The
quickest way to invert a 2×2 matrix is usually with the formula a b c d−1 = 1 ad−bc d −b −c a.) (d) (3 marks) Compute V D7V −1 and V D8V −1. (Show working.) (e) (2
marks) Find the number of possums and rats predicted by the model at the end of 7 years. Use the nearest whole numbers. (f) (2 mark) Which population eventually dies
out? In which year does this occur?
3. Five island chiefs discuss politics around a bowl of kava. The bowl is at (0,0) in (x,y) coordinates. The 5 chiefs are at (−2,0),(−1,2),(0,2),(2,1), and (2,0). This
looks vaguely like they are sitting in a semicircle. Your job is to find a ‘best fit’ circular curve of the form x2 + y2 = ax + by + c.
(a) (2 marks) Using the chiefs’ positions, write down a system of 5 linear equations in a,b,c. (b) (2 marks) Suppose the system is given in matrix form as Ax = b, with
x = a b c . What areA and b? (c) (4 marks) Compute the matrix ATA and vector ATb. (d) (2 marks) Find the solution to the normal equation ATAx = ATb. You can do
this at the Wolfram Alpha website by copying and pasting the following url into your browser
http://www.wolframalpha.com/input/?i=rowreduce{{1,1,1,1},{1,1,1,1},{1,1,1,1}}
and then changing the 1s to the values calculated in part (c). (e) (2 marks) Plot the best fit circle together with the original data. A plot of a circle can be found
at
www.wolframalpha.com/input/?i=plot+x%5E2%2By%5E2%3Dx%2By%2B1
(you need to put in the a,b,c coefficients). You can save your best fit circle as an image file (eg right click → Save image as). Then print the image and mark the chiefs’
positions on the printout.
4. Solve the following differential equations for y = y(t), with the given conditions. Show your work.
(a) (3 marks)
y0 + 2ty2 = 0, y(0) = 1.
(b) (3 marks)
t2y0 + 2ty = 3t + 1, y(1) = 1.
