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CCI Math251 Linear Algebra 2015 Assignment 4 Due Date: 10 May 2015 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Determine whether the statement is true or false: 1. A linear transformation preserves the operations of vector addition and scalar multiplication. (True). 2. If the linear transformation π : π βΆ π is both one-toone and onto, then it is an isomorphism. (True). 3. If we interchange two rows in an identity matrix, then it will not have an LU-decomposition. (True). 4. Every square matrix has an LU-decomposition. (False). 5. LU decompositions are unique. (False). 6. Simplex method is an iterative procedure for solving LPP in finite number of steps. (True). 7. The solution set of a system of linear equation is bounded if it can be enclosed by a circle. (True). - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 2 For Each Question, Choose the Correct Answer from the Multiple-Choice List. 1. One of the following matrices has no LU-decomposition 1 0 a) [0 1 0 0 0 0] 1 2 1 β1 b) [β2 β1 2 ] 2 1 0 0 1 0 c) [1 0 0] 0 0 1 1 d) [2 4 0 0 3 0] 1 2 2. Let π: π βΆ π be a linear transformation, then: a. The kernel of π is a subspace of π b. The kernel of π is a subspace of π c. The range of π is a subspace of π d. None. 3. Graphical method can be applied to solve LPP when there are only: a. One variable. b. Two variables. c. Three variables. d. None. 3 1 4. Let π: β2 βΆ β2 be the multiplication by [ ] then 4 2 π₯ π β1 ([π¦])will be equal to: a. T β 1(x, y) = (x β 2y, β 2x + 3/2y) b. T β 1(x, y) = (x β 1/2y, β 2x + 5y) c. T β 1(x, y) = (x β 1/2y, β 2x + 3/2y) d. T β 1(x, y) = (x β 1/2y, 2x + 3/2y) Solve the following questions: 1. Determine whether the function CCI Math251 Linear Algebra 2015 π: β2 βΆ β2 , where π(π₯, π¦) = (π₯ 2 , π¦ )is linear? Hint: check if this transformation preserves additivity. Solution: We have π((π₯, π¦) + (π§, π€)) = π(π₯ + π§, π¦ + π€) = ((π₯ + π§)2 , π¦ + π€) β (π₯ 2 , π¦) + (π§ 2 , π€) = π(π₯, π¦) + π(π§, π€). So, T does not preserve additivity. So, T is not linear. 2. Let T(x, y, z) = (3x β 2y + z, 2x β 3y, y β 4z). Write down the standard matrix of T, and compute T(2, β1, β1). Solution: 3 A= [2 0 β2 β3 1 1 0] β4 T(2, β1, β1) = 3 [2 0 β2 β3 1 1 7 2 0 ] [β1] = [7] 3 β4 β1 3. Find an LU-decomposition of the following matrix: A=[ 1 2 3 ] 5 Solution: 1 [ 2 π΄ = πΏπ 1 0 π’1 3 ]=[ ][ π1 1 0 5 π’2 π’3 ] 3 4 π’1 3 ] = [π π’ 5 1 1 1 [ 2 π’2 π1 π’2 + π’3 ] π’1 = 1, π’2 = 3, π1 = 2 π’3 = β1 1 So L = [ 2 1 0 ],U = [ β1 0 3 ]. 1 Note: factorization is not unique. Another one: L = [ 1 0 ] ,U = [ 1 0 1 2 3 ]. β1 4. Find the singular values of the matrix A if given the following matrix: 4 A A = [β2 β2 T 0 1 0 1 0] . 1 Solution: The characteristic equation for the matrix is: βπ3 + 6π2 β 11 π + 6. The eigenvalues are 1,2 and 3. Then the singular values of A are 1, β2,and β3. CCI Math251 Linear Algebra 2015 5. Using graphical method to solve the Linear Programming Problem. Maximize Subject to π = 6π₯ + 7π¦ 2π₯ + 3π¦ β€ 12 2π₯ + π¦ β€ 8 π₯, π¦ β₯ 0. Solution: Coordinates (0,0) (0,4) (4,0) (3,2) π = 6π₯ + 7π¦ 0 π = 28 π = 24 π = 32 The maximum value is π = 32 at the point (3,2). 5