17 9911----2121₁22= [-2value) of h is b in the plane spanned by a, and a₂?17. Let a =18. Let y =0V₂- [9] = []-28alue(s) of h is y in the plane generated by v₁ and v₂?82 and v₂ =-6-[-2]21. Let u =7Span (u, v) for all h and k., and b =19. Give a geometric description of Span {V₁, V₂} for the vectors12- [3]-9and v=-[:]h, and y20. Give a geometric description of Span {V₁, V₂} for the vectorsin Exercise 16.[²]2=23. a. Another notation for the vectorh-5[3]-3Show that- [3]. [*]kFor what22. Construct a 3 x 3 matrix A, with nonzero entries, and a vectorb in R³ such that b is not in the set spanned by the columnsof A.25. Let A =In Exercises 23 and 24, mark each statement True or False. Justifyeach answer.For what[3]b. The points in the plane corresponding tolie on a line through the origin.is inis [-4 3].[3]c. An example of a linear combination of vectors v₁ and v₂is the vector v₁.and b =andd. The solution set of the linear system whose augmentedmatrix is [a₁ a2 a3 b] is the same as the solutionset of the equation xa1 + x₂a2 + x3a3 = b.e. The set Span (u, v) is always visualized as a planethrough the origin.24 a. Any list of five real numbers is a vector in RS.b. The vector u results when a vector u - v is added to thevector V.c. The weights C₁,.... Cp in a linear combinationCIVI + + Cpvp cannot all be zero.d. When u and v are nonzero vectors, Span {(u, v) containsthe line through u and the origin.1vibn bie. Asking whether the linear system corresponding toan augmented matrix [a₁ a2 a3 b] has a solutionamounts to asking whether b is in Span {a₁, a2, a3).1 0-43 -20-2 6 31-[4columns of A by a₁, a2, a3, and let W = Span {a₁, a2, a3).Denote thea. Is b in (a,, a2. a3)? How many vectors are in {a,. a2. a3}?b. Is b in W? How many vectors are in W?c.Show that a, is in W. [Hint: Row operations are unnec-essary.]206851the set of all linear combinations of the columns of A.26. Let A =1.3 Vector Equations 33-11-2V₂ =let b =----1033a. Is b in W?b. Show that the third column of A is in W.and let W be27.A mining company has two mines. One day's operation atmine #1 produces ore that contains 20 metric tons of cop-per and 550 kilograms of silver, while one day's operationat mine #2 produces ore that contains 30 metric tons of20andcopper and 500 kilograms of silver. Let v₁ =550Then v₁ and v₂ represent the "output per day"30500of mine #1 and mine #2, respectively.a. What physical interpretation can be given to the vector5v₁ ?b. Suppose the company operates mine #1 for x₁ days andmine #2 for x₂ days. Write a vector equation whose solu-tion gives the number of days each mine should operate inorder to produce 150 tons of copper and 2825 kilogramsof silver. Do not solve the equation.c. [M] Solve the equation in (b).28.A steam plant burns two types of coal: anthracite (A) andbituminous (B). For each ton of A burned, the plant produces27.6 million Btu of heat, 3100 grams (g) of sulfur dioxide,and 250 g of particulate matter (solid-particle pollutants). Foreach ton of B burned, the plant produces 30.2 million Btu,6400 g of sulfur dioxide, and 360 g of particulate matter.a. How much heat does the steam plant produce when itburns x₁ tons of A and x₂ tons of B?b. Suppose the output of the steam plant is described bya vector that lists the amounts of heat, sulfur dioxide,and particulate matter. Express this output as a linearcombination of two vectors, assuming that the plant burnsx₁ tons of A and x₂ tons of B.c. [M] Over a certain time period, the steam plant produced162 million Btu of heat, 23,610 g of sulfur dioxide, and1623 g of particulate matter. Determine how many tonsof each type of coal the steam plant must have burned.Include a vector equation as part of your solution.V29. Let V₁...., Vk be points in R3 and suppose that forj = 1,..., k an object with mass m, is located at point v, .Physicists call such objects point masses. The total mass ofthe system of point masses ism = m₁ + ... + mk

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17 9911----2121₁22= [-2value) of h is b in the plane spanned by a, and a₂?17. Let a =18. Let y =0V₂- [9] = []-28alue(s) of h is y in the plane generated by v₁ and v₂?82 and v₂ =-6-[-2]21. Let u =7Span (u, v) for all h and k., and b =19. Give a geometric description of Span {V₁, V₂} for the vectors12- [3]-9and v=-[:]h, and y20. Give a geometric description of Span {V₁, V₂} for the vectorsin Exercise 16.[²]2=23. a. Another notation for the vectorh-5[3]-3Show that- [3]. [*]kFor what22. Construct a 3 x 3 matrix A, with nonzero entries, and a vectorb in R³ such that b is not in the set spanned by the columnsof A.25. Let A =In Exercises 23 and 24, mark each statement True or False. Justifyeach answer.For what[3]b. The points in the plane corresponding tolie on a line through the origin.is inis [-4 3].[3]c. An example of a linear combination of vectors v₁ and v₂is the vector v₁.and b =andd. The solution set of the linear system whose augmentedmatrix is [a₁ a2 a3 b] is the same as the solutionset of the equation xa1 + x₂a2 + x3a3 = b.e. The set Span (u, v) is always visualized as a planethrough the origin.24 a. Any list of five real numbers is a vector in RS.b. The vector u results when a vector u - v is added to thevector V.c. The weights C₁,.... Cp in a linear combinationCIVI + + Cpvp cannot all be zero.d. When u and v are nonzero vectors, Span {(u, v) containsthe line through u and the origin.1vibn bie. Asking whether the linear system corresponding toan augmented matrix [a₁ a2 a3 b] has a solutionamounts to asking whether b is in Span {a₁, a2, a3).1 0-43 -20-2 6 31-[4columns of A by a₁, a2, a3, and let W = Span {a₁, a2, a3).Denote thea. Is b in (a,, a2. a3)? How many vectors are in {a,. a2. a3}?b. Is b in W? How many vectors are in W?c.Show that a, is in W. [Hint: Row operations are unnec-essary.]206851the set of all linear combinations of the columns of A.26. Let A =1.3 Vector Equations 33-11-2V₂ =let b =----1033a. Is b in W?b. Show that the third column of A is in W.and let W be27.A mining company has two mines. One day's operation atmine #1 produces ore that contains 20 metric tons of cop-per and 550 kilograms of silver, while one day's operationat mine #2 produces ore that contains 30 metric tons of20andcopper and 500 kilograms of silver. Let v₁ =550Then v₁ and v₂ represent the &#34;output per day&#34;30500of mine #1 and mine #2, respectively.a. What physical interpretation can be given to the vector5v₁ ?b. Suppose the company operates mine #1 for x₁ days andmine #2 for x₂ days. Write a vector equation whose solu-tion gives the number of days each mine should operate inorder to produce 150 tons of copper and 2825 kilogramsof silver. Do not solve the equation.c. [M] Solve the equation in (b).28.A steam plant burns two types of coal: anthracite (A) andbituminous (B). For each ton of A burned, the plant produces27.6 million Btu of heat, 3100 grams (g) of sulfur dioxide,and 250 g of particulate matter (solid-particle pollutants). Foreach ton of B burned, the plant produces 30.2 million Btu,6400 g of sulfur dioxide, and 360 g of particulate matter.a. How much heat does the steam plant produce when itburns x₁ tons of A and x₂ tons of B?b. Suppose the output of the steam plant is described bya vector that lists the amounts of heat, sulfur dioxide,and particulate matter. Express this output as a linearcombination of two vectors, assuming that the plant burnsx₁ tons of A and x₂ tons of B.c. [M] Over a certain time period, the steam plant produced162 million Btu of heat, 23,610 g of sulfur dioxide, and1623 g of particulate matter. Determine how many tonsof each type of coal the steam plant must have burned.Include a vector equation as part of your solution.V29. Let V₁...., Vk be points in R3 and suppose that forj = 1,..., k an object with mass m, is located at point v, .Physicists call such objects point masses. The total mass ofthe system of point masses ism = m₁ + ... + mk

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7. Let a₁ = ------- value) of his b in the plane spanned by a and b = For what and a₂?