EE 518 : Homework #1 Due on Wednesday, September 7, 2016Problem 1Given a matrix A amd a vector α:A=1 −1 −1−1 1 10 −2 −1and α =−11−1(i) Find the determinant, trace, transpose, Row-Echelon Form and rank of A.(ii) Solve Ax=α.(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 solutionwithout a permutation matrix, then solve it using one. Both solutions will be accepted, as long as youshow the steps that you followed.)(v) Find all the eigenvalues of A and determine whether A is diagonalizable.Problem 2Determine a spanning set for nullspace(A), with A=1 2 3 51 3 4 21 4 5 −1.Problem 3Let V = <3= {(x, y, z)} for all real x, y, z. For each of the following subsets of V , determine whether or notthe 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+ 2y2= z2}Problem 4(i) Find the largest possible number of independent vectors among the following vectors:v1=1−100, v2=10−10, v3=100−1, v4=01−10, v5=001−1(ii) Find the largest possible number of independent matrices among the following matrices:A1=1 00 4, A2=−3 01 0, A3=2 11 2, A4=4 22 4Page 1 of 2EE 518 : Homework #1 Due on Wednesday, September 7, 2016Problem 5Let vector space Pk= {α0+ α1x + · · · + αk−1xk−1: αi∈ <}. Define two transformations T1: P3→ P4andT2: P3→ P2as 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 P3and determine the components of p(x) = 1+4x+2x2relative 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→ P4and T2: P3→ P2are linear transformations, determine Ker(T1), Ker(T2)and their dimensions.Problem 6This exercise is for you to get familiar with Matlab operations. For each case, write the command you usedand the result you got.A=1 −3 −1−1 0 10 −2 −5and B=9 0 −11 −1 02 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 greaterthan 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 ofelement of A at (1,3) plus the element of B at (2,1) is equal to 0, or both?Page 2 of