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7. Determine if the given sets are subspaces of the respective vector
spaces:
() = {( ) ∈ 2 | ≥ 0}; 2
() = {( ) ∈ 3 | = 0}; 3
() = {( ) ∈ 3 | = 1}; 3
In what follows, denotes the set of all × matrices.
() = All diagonal matrices of order ;
() = All 3 × 2 matrices such that 11 = 0; 32
() = [] , the set of all continuous functions : [ ] →
; <[] the set of all real-values functions defined on [ ],
() = the set of all polynomials of degree or less; < the
set of all real-valued functions : →
0
() = the set of all polynomials of degree exactly ; <
8. Determine the null space of each of¸the following matrices:
∙ ¸ ∙ ∙ ¸
2 0 1 −7 0 0
() = ;= ;=
−4 10 −3 21 0 0
9. Describe the span of each of the following sets of “vectors”:
∙ ¸ ∙ ¸
1 0 0 0
() 1 = 2 = ;
0 0 0 1
() 1 = 1 2 = 3 = 3
10. Specify a set of vectors that will exactly span each of the
following vector spaces and verify your answers::
() ; () 22 ; ()
11. Verify if the following sets of vectors will span 3 :
() 1 = (1 2 0) 2 = (3 1 0) 3 = (4 0 1);
() 1 = (4 −3 9) 2 = (2 −1 8) 3 = (6 −5 10)
12. Determine if the following sets of vectors are linearly indepen-
dent or linearly dependent:
() 1 = (−2 1) 2 = (−1 −3) 3 = (4 −2);
() 1 = (1 1 −1 2) 2 = (2 −2 0 2) 3 = (2 −8 3 −1);
() 1 = (1 −2 3 −4) 2 = (−1 3 4 2) 3 = (1 1 −2 −2)](https://image.slidesharecdn.com/sau-quantitativemethodsproblemset3-121210021135-phpapp02/85/Sau-quantitative-methods-problem-set-3-3-320.jpg)
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Sau quantitative methods problem set 3
This document contains a problem set in quantitative methods with 17 questions covering topics in linear algebra including: solving systems of linear equations using Gauss-Jordan elimination; determining the inverse of matrices; finding the null space and row/column spaces of matrices; determining if sets of vectors are linearly independent/dependent or span vector spaces; and identifying if sets of vectors form bases. The problem set is assigned by Manimay Sengupta for the Monsoon Semester 2012 at South Asian University.
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Sau quantitative methods problem set 3
- 1. 1 South Asian University Quantitative Methods Problem Set 3 Lecturer: Manimay Sengupta Monsoon Semester, 2012. 1. Use Gauss-Jordan elimination procedure to solve the following systems of linear equations: () − 21 + 2 − 3 = 4 () 1 − 22 + 33 = −2 1 + 22 + 33 = 13 − 1 + 2 − 23 = 3 31 + 3 = −1 21 − 2 + 33 = 1 () 1 − 22 + 33 = −2 () 71 + 22 − 23 − 44 + 35 = 8 −1 + 2 − 23 = 3 − 31 − 32 + 24 + 5 = −1 21 − 2 + 33 = −7 41 − 2 − 83 + 205 = 1 2. Determine, where possible, the inverse of the following matrices: ⎡ ⎤ ⎡ ⎤ ∙ ¸ 3 1 0 3 3 6 −4 −2 () = () = ⎣ −1 2 2 ⎦ () = ⎣ 0 1 2 ⎦ 5 5 5 0 −1 −2 0 0 ⎡ ⎤ ∙ ¸ 3 0 0 () = () = ⎣ 0 2 0 ⎦ 9 5 4 3. Use the inverse of the coefficient matrix to solve the following system of equations: 31 + 2 = 6 −1 + 22 + 23 = −7 51 − 3 = 10 4. Determine the conditions (if any) on 1 2 3 in order for the following systems to be consistent: () 1 − 22 + 63 = 1 () 1 + 32 − 23 = 1 −1 + 2 − 3 = 2 − 1 − 52 + 33 = 2 −31 + 2 + 83 = 3 21 − 82 + 33 = 3
- 2. 2 5. Determine if the following functions are linear transformations or not: () : 2 → 4 () : 3 → 2 (1 2 ) = (1 2 3 4 ) (1 2 3 ) = (1 2 ) 1 = 31 − 42 1 = 42 + 2 2 3 2 2 = 1 + 22 2 = 1 − 2 3 3 = 61 − 2 4 = 102 6. Examine whether the following sets are vector spaces or not: () The set = 2 with the usual definition of vector addition, and scalar multiplication defined as: (1 2 ) = (1 2 ) () The set The set = 3 with the usual definition of vector addition, and scalar multiplication defined as: (1 2 3 ) = (0 0 3 ) () The set = 2 with the usual definition of scalar multiplica- tion, and vector addition defined as: (1 2 ) + (1 2 ) = (1 + 21 2 + 2 ) (d) The set = { ∈ | 0} with addition and scalar multiplication defined as follows: + = ; = (e) The set of the points on a line through the origin in 2 with the usual addition and scalar multiplication. (f ) The set of the points on a line that does not go through the origin in 2 with the usual addition and scalar multiplication. (g) The set of the points on a plane that goes through the origin in 3 with the standard addition and scalar multiplication. (Note: The equation of a plane through the origin is + + = 0 where are given constants.)
- 3. 3 7. Determine if the given sets are subspaces of the respective vector spaces: () = {( ) ∈ 2 | ≥ 0}; 2 () = {( ) ∈ 3 | = 0}; 3 () = {( ) ∈ 3 | = 1}; 3 In what follows, denotes the set of all × matrices. () = All diagonal matrices of order ; () = All 3 × 2 matrices such that 11 = 0; 32 () = [] , the set of all continuous functions : [ ] → ; <[] the set of all real-values functions defined on [ ], () = the set of all polynomials of degree or less; < the set of all real-valued functions : → 0 () = the set of all polynomials of degree exactly ; < 8. Determine the null space of each of¸the following matrices: ∙ ¸ ∙ ∙ ¸ 2 0 1 −7 0 0 () = ;= ;= −4 10 −3 21 0 0 9. Describe the span of each of the following sets of “vectors”: ∙ ¸ ∙ ¸ 1 0 0 0 () 1 = 2 = ; 0 0 0 1 () 1 = 1 2 = 3 = 3 10. Specify a set of vectors that will exactly span each of the following vector spaces and verify your answers:: () ; () 22 ; () 11. Verify if the following sets of vectors will span 3 : () 1 = (1 2 0) 2 = (3 1 0) 3 = (4 0 1); () 1 = (4 −3 9) 2 = (2 −1 8) 3 = (6 −5 10) 12. Determine if the following sets of vectors are linearly indepen- dent or linearly dependent: () 1 = (−2 1) 2 = (−1 −3) 3 = (4 −2); () 1 = (1 1 −1 2) 2 = (2 −2 0 2) 3 = (2 −8 3 −1); () 1 = (1 −2 3 −4) 2 = (−1 3 4 2) 3 = (1 1 −2 −2)
- 4. 4 13. Determine if the following sets of vectors are linearly indepen- dent or linearly dependent: ∙ ¸ ∙ ¸ ∙ ¸ 1 0 0 0 0 1 0 0 0 () 1 = 2 = 3 = ; 0 0 0 0 0 0 0 1 0 ∙ ¸ ∙ ¸ 1 2 4 1 () 1 = 2 = ; 0 −1 0 −3 () 1 = 1 2 = 3 = 2 in 2 ; () 1 = 22 − + 7 2 = 2 + 4 + 2 3 = 2 − 2 + 4 in 2 14. Examine if each of the following sets of vectors will be a basis for 3 : () 1 = (1 0 0) 2 = (0 1 0) 3 = (0 0 1); () 1 = (1 −2 1) 2 = (2 −1 3) 3 = (5 −3 −1); () 1 = (1 1 0) 2 = (−1 0 0) 15. Examine if each of the following sets of vectors will form a basis for the indicated vector space: () 0 = 1 2 = 3 = 2 = ; ∙ ¸ ∙ ¸ 1 0 0 0 () 1 = 2 = 0 0 1 0 ∙ ¸ ∙ ¸ 0 1 0 0 3 = 4 = 0 0 0 1 16. Determine the basis and dimension of the null space of the following matrices: ⎡ ⎤ ⎡ ⎤ 7 2 −2 −4 3 2 −4 1 2 −2 −3 () = ⎣ −3 −3 0 2 1 ⎦ ; () = ⎣ −1 2 0 0 1 −1 ⎦ 4 −1 −8 0 20 10 −4 −2 4 −2 4 17. Find the row and the column spaces of the following matrix: ⎡ ⎤ 1 5 −2 3 5 ⎢ 0 0 1 −1 0 ⎥ =⎢ ⎣ 0 0 0 ⎥ 0 1 ⎦ 0 0 0 0 0
- 5. 5 18. Find a basis for the row and the column spaces of the matrices and in Problem 16, and thus determine the rank of these matrices. 19. Find a basis for the row space, the column space and the null space of the following matrices. Determine the rank and nullity of the matrices. ⎡ ⎤ −1 2 −1 5 6 ⎡ ⎤ ⎢ 4 −4 −4 −12 −8 ⎥ 6 −3 () = ⎢⎣ 2 ⎥ ; () = ⎣ −2 3 ⎦ 0 −6 −2 4 ⎦ −8 4 −3 1 7 −2 12 20. For each of the following matrices, determine the eigenvectors and a basis for the eigenspace of these matrices corresponding to each of their eigenvalues: ∙ ¸ ∙ ¸ 6 16 7 −1 () = ; () = −1 −4 4 3 21. For each of the following matrices, determine the eigenvectors and a basis for the eigenspace of these matrices corresponding to each of their eigenvalues: ⎡ ⎤ ⎡ ⎤ 4 0 1 6 3 −8 () = ⎣ −1 −6 −2 ⎦ ; () = ⎣ 0 −2 0 ⎦ ; 5 0 0 1 0 −3 ⎡ ⎤ ⎡ ⎤ 0 1 1 4 0 −1 () = ⎣ 1 0 1 ⎦ ; () = ⎣ 0 3 0 ⎦ 1 1 0 1 0 2 22. Write the following quadratic forms in matrix form () = 0 where is a symmetric matrix: (a) ( ) = 2 + 2 + 2 ; () ( ) = 2 + + 2 ; () (1 2 3 ) = 32 − 21 2 + 31 3 + 2 − 42 3 + 32 1 3
- 6. 6 23. Classify the following quadratic forms, whether positive defi- nite, positive semidefinite etc.: () (1 2 ) = 2 + 82 ; () (1 2 3 ) = 2 + 82 ; 1 2 2 3 () (1 2 3 ) = 52 + 21 3 + 22 + 22 3 + 42 ; 1 2 3 () (1 2 3 ) = −31 + 21 2 − 2 + 42 3 − 82 2 2 3