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Problems For Problems 1-14, determine whether the given set S of vectors is closed under addition and closed under scalar mul- tiplication. In each case, take the set of scalars to be the set of all real numbers. 1. The set SQ of all rational numbers.¹

Problems For Problems 1-14, determine whether the given set S of vectors is closed under addition and closed under scalar mul- tiplication. In each case, take the set of scalars to be the set of all real numbers. 1. The set SQ of all rational numbers.¹

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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and scalar multiplication, forms a vector space. (i) The set of positive real numbers, with the usual op- erations of addition and scalar multiplication, forms a vector space. (j) The set of nonnegative real numbers (i.e., {x € R : x ≥ 0}), with the usual operations of addition and scalar multiplication, forms a vector space. Problems For Problems 1-14, determine whether the given set S of vectors is closed under addition and closed under scalar mul- tiplication. In each case, take the set of scalars to be the set of all real numbers. 1. The set S := Q of all rational numbers. ¹ 2. The set S = Un(R) of all upper triangular n × n matrices with real elements. 3. The set S := {A € M₂ (R): A is upper or lower triangular}. 4. The set S := {ao + a₁x + a₂x² : ao + a₁ + a2 = 0}. 5. The set S := {ao + a₁x + a2x² : ao + a₁ + a₂ = 1}. 6. The set S of all solutions to the differential equation y' + 3y 6x³ + 5. (Do not solve the differential equation.) = 7. The set S of all solutions to the differential equation y' + 3y = 0. (Do not solve the differential equation.) 14. The set S of all polynomials of the forma + bx³ + cx4, where a, b, c ER. = 15. We have defined the set R² {(x, y) : x, y ≤ R}, together with the addition and scalar multiplication operations as follows: (x1, y₁) + (x2, Y2) = (x1 + x2, y1 + y2), k(x₁, y₁) = (kxı, ky₁). Give a complete verification that each of the vector space axioms is satisfied. = 16. Determine the zero vector in the vector space V M4x2 (R), and write down a general element A in V along with its additive inverse - A. 17. Generalize the previous exercise to find the zero vec- tor and the additive inverse of a general element of Mmxn (R). = 18. Determine the zero vector in the vector space V P3 (R), and write down a general element p(x) in V along with its additive inverse -p(x). 19. Generalize the previous exercise to find the zero vec- tor and the additive inverse of a general element of Pn (R). 20. On R+, the set of positive real numbers, define the op- erations of addition, , and scalar multiplication, O, as follows: x + y = = xy COX = = xc.
Transcribed Image Text:and scalar multiplication, forms a vector space. (i) The set of positive real numbers, with the usual op- erations of addition and scalar multiplication, forms a vector space. (j) The set of nonnegative real numbers (i.e., {x € R : x ≥ 0}), with the usual operations of addition and scalar multiplication, forms a vector space. Problems For Problems 1-14, determine whether the given set S of vectors is closed under addition and closed under scalar mul- tiplication. In each case, take the set of scalars to be the set of all real numbers. 1. The set S := Q of all rational numbers. ¹ 2. The set S = Un(R) of all upper triangular n × n matrices with real elements. 3. The set S := {A € M₂ (R): A is upper or lower triangular}. 4. The set S := {ao + a₁x + a₂x² : ao + a₁ + a2 = 0}. 5. The set S := {ao + a₁x + a2x² : ao + a₁ + a₂ = 1}. 6. The set S of all solutions to the differential equation y' + 3y 6x³ + 5. (Do not solve the differential equation.) = 7. The set S of all solutions to the differential equation y' + 3y = 0. (Do not solve the differential equation.) 14. The set S of all polynomials of the forma + bx³ + cx4, where a, b, c ER. = 15. We have defined the set R² {(x, y) : x, y ≤ R}, together with the addition and scalar multiplication operations as follows: (x1, y₁) + (x2, Y2) = (x1 + x2, y1 + y2), k(x₁, y₁) = (kxı, ky₁). Give a complete verification that each of the vector space axioms is satisfied. = 16. Determine the zero vector in the vector space V M4x2 (R), and write down a general element A in V along with its additive inverse - A. 17. Generalize the previous exercise to find the zero vec- tor and the additive inverse of a general element of Mmxn (R). = 18. Determine the zero vector in the vector space V P3 (R), and write down a general element p(x) in V along with its additive inverse -p(x). 19. Generalize the previous exercise to find the zero vec- tor and the additive inverse of a general element of Pn (R). 20. On R+, the set of positive real numbers, define the op- erations of addition, , and scalar multiplication, O, as follows: x + y = = xy COX = = xc.
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