EE 518 Homework 1 Due on Wednesday September 7 2016 Problem 1 Given a matrix A amd a vector 1 A 1 0 1 1 1 1 1 and 1 1 2 1 i Find the determinant trace transpose Row Echelon Form and rank of A ii Solve Ax iii Find the nullspace of A iv Find a lower triangular L and an upper triangular U so that A LU Tip If you cannot get a solution without a permutation matrix then solve it using one Both solutions will be accepted as long as you show the steps that you followed v Find all the eigenvalues of A and determine whether A is diagonalizable Problem 2 1 Determine a spanning set for nullspace A with A 1 1 2 3 4 3 4 5 5 2 1 Problem 3 Let V 3 x y z for all real x y z For each of the following subsets of V determine whether or not the subset is a subspace of V If it is not a subspace show how one of the vector space requirement fails i S1 all x y z with x 2y 3z ii S2 all x y z with x 2y z iii S3 all x y z with x y 3z 5 iv S4 all x y z with x2 2y 2 z 2 Problem 4 i Find the largest possible number of independent vectors among the following vectors 1 1 1 0 0 1 0 0 1 0 v1 0 v2 1 v3 0 v4 1 v5 1 0 0 1 0 1 ii Find the largest possible number of independent matrices among the following matrices 1 A1 0 0 3 A2 4 1 0 2 A3 0 1 1 4 A4 2 2 2 4 Page 1 of 2 EE 518 Homework 1 Due on Wednesday September 7 2016 Problem 5 Let vector space Pk 0 1 x k 1 xk 1 i Define two transformations T1 P3 P4 and T2 P3 P2 as follows For every polynomial p x in P3 T1 p x xp x and T2 p x p0 x dp x dx Let p1 x 1 p2 x x x2 p3 x x x2 i Show that p1 x p2 x p3 x form a basis for P3 and determine the components of p x 1 4x 2x2 relative to this basis ii Is T1 p1 x T1 p2 x T1 p3 x a spanning set for P4 Is T2 p1 x T2 p2 x T2 p3 x a spanning set for P2 iii Show that T1 P3 P4 and T2 P3 P2 are linear transformations determine Ker T1 Ker T2 and their dimensions Problem 6 This exercise is for you to get familiar with Matlab operations For each case write the command you used and the result you got 1 A 1 0 3 1 9 0 1 and B 1 2 5 2 0 1 1 0 5 1 i Find the transpose of A ii Calculate A B iii Raise each element of B to the power of 3 iv Produce a 3 by 3 matrix C whose elements should have value 1 if the respective element of A is greater than or equal to the respective element of B and 0 otherwise v Is it true that either the element of A at 2 3 is greater than the element of B at 3 1 or the sum of element of A at 1 3 plus the element of B at 2 1 is equal to 0 or both Page 2 of 2