32 CHAPTER 1 Linear Equations in Linear Algebra mod 1. u = mariw 5 101 sigis 2. u= V= 5. x₁ -1 + x₂ 5 1.3 EXERCISES In Exercises 1 and 2, compute u + vand u-2v. 201016 d с Tonalq tadi 2. VEROR -2v 4 0 7. Vectors a, b, c, and d 8. Vectors w, x, y, and z [2]-V = [-2] In Exercises 3 and 4, display the following vectors using arrowsector bis 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 ya 4. u and v as in Exercise 2 ebas,10th m Indys.cus In Exercises 5 and 6, write a system of equations that is equivalent op to the given vector equation. b PRACTICE PROBLEMS 1. Prove that u + v = v +u for any u and v in R". For what value(s) of h will y be in Span{V₁, V2, V3} if 5 [1] [3] Jos snil bas asigillum ulega woda 9. 27phogoleo la mwalno. lo atinu Istovos gnisubong to Janojodi -V -u 6. X₁ * [3] + * [$]+ * [ 6 ] - [8] *[-]-[8] T x₂ x3 14001 Fimo noin 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? (basdrovo T 1 V₁ = chi Bna 10 u 3. Let W₁, W2, W3, u, and v be vectors in R". Suppose the vectors u and v are in Span (W₁, W2, W3}. Show that u + v is also in Span (W₁, W2, W3}. [Hint: The solution to Practice Problem 3 requires the use of the definition of the span of a set of vectors It is useful to review this definition on Page 30 before starting this exercise.] V bris -2 In Exercises 9 and 10, write a vector equation that is equivalent to ndu-2 riferols the given system of equations. W du 10t09 -5 GeometMeDescriptio VA as. 2v V2 = z 4-2001 ou 8 10 dnow inalloli to drow olmx₂ + 5x3 = 0 4x₁ + 6x2x3 = 0 amppr -x₁ + 3x₂8x3 = 0 MOITO-31 V3 = 11. a₁ = alinoge 0912 In Exercises 11 and 12, determine if b is a linear combination of a₁, a2, and a3. 12. a₁ = -2 Glytoim 13. A = 1 14. A = -----0---- = 5 , a3 = 5 5 H-0--0-H -2,a₂ = 1 , a3 = -6 8 15. V₁ = [1] Γ the 27vector-57 ,b= 2 Tricht 6 16. V₁ = 1 -4 2 0 3 5 b= -2008 -4 In Exercises 13 and 14, determine if b is a linear combination of the vectors formed from the columns of the matrix A. -2 -6 7,b= 5 0 3 -2 and y = 1 DES 10. 4x1 + x2 + 3x3 = 9 x₁7x₂2x3 = 2 8x1 + 6x25x3 = 15 hay bill 252 jalshatam vol-6² = 3 0 . men -4 3 [37ST 0 2 , V₂ = -2 0 3 8 studimon 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. b = 3 -7 -3 2 6 1