Use the accompanying figure to write each vector listed as a linear combination of u and v. Vectors c, x, y, and z Write c as a linear combination of u and v. (Type integers or decimals.) Write x as a linear combination of u and v. (Type integers or decimals.) Write y as a linear combination of u and v. (Type integers or decimals.) Write z as a linear combination of u and v.
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Work With Experts to Reach at Correct Answers![Let u = [-10 -8 -7], A = [6 7 16 0 2 -4 1 2 1]. Is u in the subset of R^3 spanned by the columns of A? Why or why not? Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or decimal for each matrix element.) A. No, the reduced row echelon form of the augmented matrix is which is an inconsistent system. B. Yes, multiplying A by the vector writes u as a linear combination of the columns of A.](https://www.doubtrix.com/js/ckeditor/filemanager/connectors/php/editor/1700960259-Q6.JPG)
![5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]](https://www.doubtrix.com/uploads/editor/6372428458LgmZXNxqka.png)
![5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]](https://www.doubtrix.com/uploads/editor/6896791134NPlCNIwqTC.png)
![5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]](https://www.doubtrix.com/uploads/editor/1401586756DoSHMhHJpq.png)
![5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]](https://www.doubtrix.com/uploads/editor/9702922078OiTwyPXqkE.png)
