Prof. E. Berlekamp September 21, 1999 Math 110 QUIZ Summary: Problem 1: LU factorization 30 points Problem 2: Binary erasure correction 30 points (10 pts/part) Problem 3: Basis and Dimension 40 points (10 pts/part) Problem 4: Rudimentary operator algebra 20 points 1) Without doing any permutations on the following real matrix, factor it into LU, where L is lower-triangular, and U is upper triangular. L should have all ones on its main diagonal. 2 1 0 0 0 3 2 1 0 0 0 3 2 1 0 0 0 3 2 1 0 0 0 3 2 2) The "Hamming" code is a binary vector space, defined as the nullspace of the following 4x8 binary matrix: 0 0 0 0 1 1 1 1 H = 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 1 0 The vectors in this code are called codewords. Any codeword is a possible message. Transmission noise might cause some bits of a message to be "erased", and replaced with "?"s. 2a) Suppose the received word is x0 0 x1 ? x2 1 x3 = 1 x4 ? x5 0 x6 ? x7 1 If possible, using any method you like, find the values of the bits in the three erased locations: x1, x4, and x6. 2b) If possible, find a set of 4 locations which, if erased, can be uniquely decoded. 2c) If possible, find a set of 4 locations which, if erased, can NOT be uniquely decoded. 3) Each part of this problem defines a vector space or subspace. Determine its dimension, and exhibit a basis. 3a) The space of all vectors in R^4 whose components add to zero 3b) The nullspace of the following real matrix: 1 2 0 1 2 1 U = 0 0 2 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3c) The space of all symmetric 3x3 matrices. 3d) The subspace of R^infinity which consists of all arithmetic progressions, including (1/2, 3/2, 5/2, 7/2, 9/2,... ) ( 10, 7, 4, 1, -2,... ) (1 7, 18, 19, 20, 21,... ) 4) Let I denote the square nxn identity matrix, and let J denote the nxn matrix which has "1" is each of its n^2 components: 1 1 1 ... 1 1 1 1 ... 1 J = 1 1 1 ... 1 ..... ... . 1 1 1 ... 1 Let b be a scalar. In terms of b and n, determine the value of another scalar, c, such that (I + bJ) is the inverse of (I + cJ).