Question

1. Write R' = {(x, y) |X, Y ER} to represent the set of all 1x2 row vectors of real numbers. This is the standard Euclidean plane you all know and love. If such a row vector is multiplied on the right by a 2x2 matrix, then the output is again in R"; such matrices are called linear transformations. 1. Find a linear transformation that rotates the plane R by a radians. That is, find a matrix T such that (x, y) = (-X, -y). "Verify that the function (x, y) (-X, -y) really is rotation by . Hint: You can make a system of four linear equations in the four unknown entries of T by considering what T does to the row vectors (1,0) and (0,1). 2. Find a linear transformation that rotates the plane by /2 radians clockwise. 3. What about rotation clockwise by an arbitrary angle 8? (You'll have to remember basic trig for this one) 2. Recall the Fibonacci numbers: 0,1, 1, 2, 3, 5, 8, 13, 21, 34, .... If we denote the fo = 0 and fi = 1, then we can define the nth Fibonacci as fn = fn-2 +fn-1- To find f, requires roughly n arithmetic operations (1.e., is O(n)). Matrices will allow us to not only compute fr faster, but to even find a formula (later). For now, write x, = "n+ ); these are column vectors of consecutive Fibonacci numbers. So I fn , * = () x = ( ) * = (*) * = (?). « = (3) .ete. First, find a transition matrix, i.e, a matrix T such that Tx, = Xn+1. Second, find a matrix M such that Mxo = xn. (*) Show that finding fn with the matrices above is O(log, n). Hint: Start with n = 2 and describe how many multiplications of 2x2 matrices are needed to compute M above. Generalize to arbitrary n by writing n in binary. (*) There are n citizens living in City Z. Their main occupation is forming various clubs, which at some point started threatening the very survival of the city. In order to limit the number of clubs, the city council decreed the following innocent-looking rules: • Each club must have an odd number of members. • Every two clubs must have an even number of members in common. Show that these rules are enough to limit the number of clubs to at most n. Hint: Start by creating a matrix M whose entries encode which citizen is in which club. Consider the entries of M to be in the field F. What can you say about the rank of M? Compute MMT. What do the entries of this matrix mean? What do the rules of the council imply about MM'? What does this imply about the rank of M? (*) Find matrices A such that 1. A + 0, but A2 = 0 2. A2 € 0, but A3 = 0 Make a conjecture about matrices satisfying A"-'70 but A" = 0.

Answer 1

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t-02 T( ) m 0+02) =TCODO+02) a ! cOA0tom9(b-1m0) B1no CODO -0/10 Lob0tyAmo, -81m0 t op0J TIM.9)= on