In Exercises 3 and 4, display the following vectors using arrows on an xy-graph: u, v, -v, -2v, u + v, u-v, and u-2v. Notice that u - v is the vertex of a parallelogram whose other vertices are u, 0, and -v. 3. u and v as in Exercise 1 4. u and v as in Exercise 2 In Exercises 5 and 6, write a system of equations that is equivalent to the given vector equation. 5. X₁ -1 + X₂ 4 0 6. X₁ *[¯3 ] + x[$]+*[_!]-[8] Use the accompanying figure to write each vector listed in Exer- cises 7 and 8 as a linear combination of u and v. Is every vector in R² a linear combination of u and v? In Exercises 11 and 12, determine if b is a linear combination of a₁, a₂, and a. 11. a₁ = 12. a₁ = 0 -------] -6 8 = ----0--0- = 13. A = 8 ,b= In Exercises 13 and 14, determine if b is a linear combination of the vectors formed from the columns of the matrix A. 1-4 2 -40-0 3 5 ,b= -2 8-4/n14. A = 1 -2 -6 0 3 7 -2 5 In Exercises 15 and 16, list five vectors in Span {V₁, V₂). For each vector, show the weights on v₁ and v₂ used to generate the vector and list the three entries of the vector. Do not make a sketch. 15. V₁ = -0--0 -6 3 -19].M 2 16. V₁ = HE] b V₂ -5 3/n19. Give a geometric description of Span (V₁, V₂} for the vectors -[3] V₁ = 2 and V₂ 1 20. Give a geometric description of Span (V₁, V₂) for the vectors in Exercise 16. 21. Let u = +-=[-²] - [3]. and v= Span (u, v) for all h and k. Show that is in 22. Construct a 3 x 3 matrix A, with nonzero entries, and a vector b in R³ such that b is not in the set spanned by the columns of A. In Exercises 23 and 24, mark each statement True or False. Justify each answer. 23. a. Another notation for the vector [] is [-4_3]. b. The points in the plane corresponding to [-] [2] lie on a line through the origin. and c. An example of a linear combination of vectors V₁ and V₂ is the vector V₁- d. The solution set of the linear system whose augmented matrix is [a₁ a2 a3 b] is the same as the solution set of the equation x₁α₁ + X₂ả₂ + X3â3 = b. e. The set Span (u, v) is always visualized as a plane through the origin. 24. a. Any list of five real numbers is a vector in R³. b. The vector u results when a vector u - v is added to the vector v. c. The weights C₁,..., Cp in a linear combination C₁V₁ ++ cpv, cannot all be zero. d. When u and v are nonzero vectors, Span {(u, v) contains the line through u and the origin. e. Asking whether the linear system corresponding to an augmented matrix [a a2 a b] has a solution amounts to asking whether b is in Span (a₁, a₂, a3)./n2 10 3 26. Let A = -1 5, let b = 1 1 3 the set of all linear combinations of the columns of A. a. Is b in W? b. Show that the third column of A is in W. 0 6 8 -2 and let W be

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