U N I V E R S I T Y O F W I S C O N S I N - M A D I S O NComputer Sciences DepartmentCS513, Spring 99Prof. RonMidterm ExamMy name is: ................Answer every question below. Write your answer into your blue book. Use a new page foreach question. You are not allowed to use books or notes. A calculator of any type is permitted,provided that it is not preprogrammed with code relevant to cs513. Be brief and to the point withyour answers. “Yes/no” answers carry “no” credit, unless reasoning is provided. Turn in your examsheet together with your blue book.(1) (30 points, 5+15+10) You are given the following 2 × 2 matrix AA =2 −10√3(a) Find the spectrum of this matrix.(b) Find the 2-, ∞- and 1-norm of this matrix.(c) Find the 2-condition number of A (if possible, without computing the inverse of A). State anyresult from class that you use here.(2) (25 points, 10+15)(a) Let v be the column vector defined byv0:= ( 3 2 4 4 ) .Find a Householder matrix H such that(Hv)0= ( 3 a 0 0 ) ,with a some number (that you might choose to suit your needs.)(b) QR-factor the matrix5 3 10 2 00 4 00 4 0.(3) (20 points, 10+10)(a) Describe briefly an efficient algorithm for computing the determinant of a square n ×n matrix.What is the complexity of your algorithm?(b) Use (a) in order to find the determinant of the matrixA =+4 −2 +2−2 +2 −2+2 −2 +3(4) (20 points, 5+10+5)(a) A is a square matrix. Define the notion: ‘A is positive definite’.(b) State two conditions, each of which is equivalent, for a symmetric matrix A, to the positivedefiniteness of that matrix.(c) Check whether the matrix in the previous problem (3(b)) is positive definite.(5) (25 points =5+20 points) Let V be the linear span of the vectors111,100.You are asked to find a vector v in V which is ‘as close as possible’ to the vectoru =124.(a) Define rigorously the notion of ‘as close as possible’. (If there are several possible definitions,choose the one that will allow you to solve (b) below.)(b) Find that ‘closest’ vector