Solve #5, SHow all of your work and give explanations. POST PICTURES OF YOUR WORK 296 CHAPTER 4Vector Spaces• Know what information the Wronskian does (and doesnot) give about the linear dependence or linear inde-pendence of a set of functions on an interval I.True-False ReviewFor Questions (a)-(i), decide if the given statement is trueor false, and give a brief justification for your answer. If true,you can quote a relevant definition or theorem from the text.If false, provide an example, illustration, or brief explanationof why the statement is false.(a) Every vector space V possesses a unique minimalspanning set.(b) The set of column vectors of a 5 x 7 matrix A must belinearly dependent.(c) The set of column vectors of a 7 x 5 matrix A must belinearly independent.(d) Any nonempty subset of a linearly independent set ofvectors is linearly independent.(e) If the Wronskian of a set of functions is nonzero atsome point xo in an interval I, then the set of func-tions is linearly independent.(f) If it is possible to express one of the vectors in a setS as a linear combination of the others, then S is alinearly dependent set.(g) If a set of vectors S in a vector space V contains alinearly dependent subset, then S is itself a linearlydependent set.(h) A set of three vectors in a vector space V is linearly de-pendent if and only if all three vectors are proportionalto one another.(i) If the Wronskian of a set of functions is identicallyzero at every point of an interval I, then the set offunctions is linearly dependent.ProblemsFor Problems 1-10, determine whether the given set ofvectors is linearly independent or linearly dependent inR". In the case of linear dependence, find a dependencyrelationship.1. {(3, 6, 9)}.2. {(1, 1), (1, 1)}.3. {(2, 1), (3, 2), (0, 1)).4. {(1, 1, 0), (0, 1, −1), (1, 1, 1)}.5. {(1, 2, 3), (1, −1, 2), (1, −4, 1)}.6. {(2, 4, 6), (3, -6, 9)}.7. {(1, 1, 2), (2, 1, 0)).8. {(1, 1, 2), (0, 2, 1), (3, 1, 2), (-1,-1, 1)}.9. {(1, 1, 2, 3), (2, −1, 1, −1), (−1, 1, 1, 1)}.10. {(2, 1, 0, 1), (1, 0, −1, 2), (0, 3, 1, 2),(-1, 1, 2, 1)).11. Let v₁ = (1, 2, 3), v₂ = (4, 5, 6), V3 = (7, 8, 9). De-termine whether {V1, V2, V3} is linearly independent inR³. Describespan{V1, V2, V3}geometrically.12. Consider the vectors v₁ = (2, -1, 5), v₂ = (1, 3, –4),V3 = (-3, 9, 12) in R³.(a) Show that {V₁, V2, V3} is linearly dependent.(b) Is v₁ € span{v2, v3}? Draw a picture illustratingyour answer.13. Determine all values of the constant k for which thevectors (1, 1, k), (0, 2, k) and (1, k, 6) are linearly de-pendent in R³.For Problems 14-15, determine all values of the constant kfor which the given set of vectors is linearly independentin R4.14. {(1, 0, 1, k), (-1, 0, k, 1), (2, 0, 1, 3)}.15. {(1, 1, 0, 1), (1, k, 1, 1), (2, 1, k, 1), (−1, 1, 1, k)}.For Problems 16-18, determine whether the given set of vec-tors is linearly independent in M₂ (R).- [6₂1]^² - [6 ]-^² - [84],-, A3 =0116. A₁ =- [34]^2-[13]=17. A₁ =- [12]·^² = [ 2² ]-^² = [3₂²], A₂, A318. A₁ =

Question

Solve #5, SHow all of your work and give explanations. POST PICTURES OF YOUR WORK 296 CHAPTER 4Vector Spaces• Know what information the Wronskian does (and doesnot) give about the linear dependence or linear inde-pendence of a set of functions on an interval I.True-False ReviewFor Questions (a)-(i), decide if the given statement is trueor false, and give a brief justification for your answer. If true,you can quote a relevant definition or theorem from the text.If false, provide an example, illustration, or brief explanationof why the statement is false.(a) Every vector space V possesses a unique minimalspanning set.(b) The set of column vectors of a 5 x 7 matrix A must belinearly dependent.(c) The set of column vectors of a 7 x 5 matrix A must belinearly independent.(d) Any nonempty subset of a linearly independent set ofvectors is linearly independent.(e) If the Wronskian of a set of functions is nonzero atsome point xo in an interval I, then the set of func-tions is linearly independent.(f) If it is possible to express one of the vectors in a setS as a linear combination of the others, then S is alinearly dependent set.(g) If a set of vectors S in a vector space V contains alinearly dependent subset, then S is itself a linearlydependent set.(h) A set of three vectors in a vector space V is linearly de-pendent if and only if all three vectors are proportionalto one another.(i) If the Wronskian of a set of functions is identicallyzero at every point of an interval I, then the set offunctions is linearly dependent.ProblemsFor Problems 1-10, determine whether the given set ofvectors is linearly independent or linearly dependent inR&#34;. In the case of linear dependence, find a dependencyrelationship.1. {(3, 6, 9)}.2. {(1, 1), (1, 1)}.3. {(2, 1), (3, 2), (0, 1)).4. {(1, 1, 0), (0, 1, −1), (1, 1, 1)}.5. {(1, 2, 3), (1, −1, 2), (1, −4, 1)}.6. {(2, 4, 6), (3, -6, 9)}.7. {(1, 1, 2), (2, 1, 0)).8. {(1, 1, 2), (0, 2, 1), (3, 1, 2), (-1,-1, 1)}.9. {(1, 1, 2, 3), (2, −1, 1, −1), (−1, 1, 1, 1)}.10. {(2, 1, 0, 1), (1, 0, −1, 2), (0, 3, 1, 2),(-1, 1, 2, 1)).11. Let v₁ = (1, 2, 3), v₂ = (4, 5, 6), V3 = (7, 8, 9). De-termine whether {V1, V2, V3} is linearly independent inR³. Describespan{V1, V2, V3}geometrically.12. Consider the vectors v₁ = (2, -1, 5), v₂ = (1, 3, –4),V3 = (-3, 9, 12) in R³.(a) Show that {V₁, V2, V3} is linearly dependent.(b) Is v₁ € span{v2, v3}? Draw a picture illustratingyour answer.13. Determine all values of the constant k for which thevectors (1, 1, k), (0, 2, k) and (1, k, 6) are linearly de-pendent in R³.For Problems 14-15, determine all values of the constant kfor which the given set of vectors is linearly independentin R4.14. {(1, 0, 1, k), (-1, 0, k, 1), (2, 0, 1, 3)}.15. {(1, 1, 0, 1), (1, k, 1, 1), (2, 1, k, 1), (−1, 1, 1, k)}.For Problems 16-18, determine whether the given set of vec-tors is linearly independent in M₂ (R).- [6₂1]^² - [6 ]-^² - [84],-, A3 =0116. A₁ =- [34]^2-[13]=17. A₁ =- [12]·^² = [ 2² ]-^² = [3₂²], A₂, A318. A₁ =

Accepted Answer

Answered: Image /qna-images/answer/8e023838-00cd-45d1-a987-566198177d93.jpg

5. {(1, 2, 3), (1, −1, 2), (1, –4, 1)}.