Problem 1PP: Let A = [152031954817], P = [3204], and b = [790]. It can be shown that p is a solution of Ax = b.... Problem 2PP: Let A = [2531], u = [41], and v = [35]. Verify Theorem 5(a) in case by computing A(u + v) and Au +... Problem 3PP: Construct a 3 3 matrix A and vectors b and c in 3 so that Ax = b has a solution, but Ax = c does... Problem 1E: Compute the products in Exercises 1-4 using (a) the definition, as in Example 1, and (b) the... Problem 2E: Compute the products in Exercises 1—4 using (a) the definition, as in Example 1, and (b) the... Problem 3E: Compute the products in Exercises 1-4 using (a) the definition, as in Example 1, and (b) the... Problem 4E: Compute the products in Exercises 1—4 using (a) the definition, as in Example 1, and (b) the... Problem 5E: In Exercises 5-8, use the definition of Ax to write the matrix equation as a vector equation, or... Problem 6E: In Exercises 5-8, use the definition of Ax to write the matrix equation as a vector equation, or... Problem 7E: In Exercises 5-8, use the definition of Ax to write the matrix equation as a vector equation, or... Problem 8E: In Exercises 5-8, use the definition of Ax to write the matrix equation as a vector equation, or... Problem 9E: In Exercises 9 and 10, write the system first as a vector equation and then as a matrix equation. 9.... Problem 10E: In Exercises 9 and 10, write the system first as a vector equation and then as a matrix equation.... Problem 11E: Given A and b in Exercises 11 and 12, write the augmented matrix for the linear system that... Problem 12E: Given A and b in Exercises 11 and 12, write the augmented matrix for the linear system that... Problem 13E: Let u=044 and A=352611. Is u in the plane in R3 spanned by the columns of A? (See the figure.) Why... Problem 14E: Let u = [232] and A = [587011130]. Is u in the subset of 3 spanned by the columns of A? Why or why... Problem 15E: Let A = [2163] and b = [b1b2]. Show that the equation Ax = b does not have a solution for all... Problem 16E: Repeat Exercise 15: A = [134326518], b = [b1b2b3]. 15. Let A = [2163] and b = [b1b2]. Show that the... Problem 17E: Exercises 17-20 refer to the matrices A and B below. Make appropriate calculations that justify your... Problem 18E: Exercises 17-20 refer to the matrices A and B below. Make appropriate calculations that justify your... Problem 19E: Exercises 17-20 refer to the matrices A and B below. Make appropriate calculations that justify your... Problem 20E: Exercises 17-20 refer to the matrices A and B below. Make appropriate calculations that justify your... Problem 21E: Let v1 = [1010], v2 = [0101], v3 = [1001]. Does {v1, v2, v3} span 4? Why or why not? Problem 22E: Let v1 = [002], v2 = [038], v3 = [415]. Does {v1, v2, v3} span 3? Why or why not? Problem 23E: In Exercises 23—34, mark each statement True or False (T/F). Justify each answer. 23. (T/F) The... Problem 24E: In Exercises 23—34, mark each statement True or False (T/F). Justify each answer. 24. (T/F) Every... Problem 25E: In Exercises 23—34, mark each statement True or False (T/F). Justify each answer. 25. (T/F) If the... Problem 26E: In Exercises 23—34, mark each statement True or False (T/F). Justify each answer. 26. (T/F) A... Problem 27E: In Exercises 23—34, mark each statement True or False (T/F). Justify each answer. 27. (T/F) The... Problem 28E: In Exercises 23—34, mark each statement True or False (T/F). Justify each answer. 28. (T/F) If A... Problem 29E: In Exercises 23—34, mark each statement True or False (T/F). Justify each answer. 29. (T/F) The... Problem 30E: In Exercises 23—34, mark each statement True or False (T/F). Justify each answer. 30. (T/F) Any... Problem 31E: In Exercises 23—34, mark each statement True or False (T/F). Justify each answer. 31. (T/F) If the... Problem 32E: In Exercises 23—34, mark each statement True or False (T/F). Justify each answer. 32. (T/F) The... Problem 33E: In Exercises 23—34, mark each statement True or False (T/F). Justify each answer. 33. (T/F) If A... Problem 34E: In Exercises 23—34, mark each statement True or False (T/F). Justify each answer. 34. (T/F) If the... Problem 35E: Note that [431525623][312]=[7310]. Use this fact (and no row operations) to find scalars c1, c2, c3... Problem 36E: Let u = [725], v = [313], and w = [610]. It can be shown that 3u 5v w = 0. Use this fact (and no... Problem 37E: Let q1, q2, q3, and v represent vectors in 5, and let x1, x2, and x3 denote scalars. Write the... Problem 38E: Rewrite the (numerical) matrix equation below in symbolic form as a vector equation, using symbols... Problem 39E: Construct a 3 3 matrix, not in echelon form, whose columns span 3. Show that the matrix you... Problem 40E: Construct a 3 3 matrix, not in echelon form, whose columns do not span 3. Show that the matrix you... Problem 41E: Let A be a 3 2 matrix. Explain why the equation Ax = b cannot be consistent for all b in 3.... Problem 42E: Could a set of three vectors in 4 span all of 4? Explain. What about n vectors in m when n is less... Problem 43E: Suppose A is a 4 3 matrix and b is a vector in 4 with the property that Ax = b has a unique... Problem 44E: Suppose A is a 3 3 matrix and b is a vector in 3 with the property that Ax = b has a unique... Problem 45E: Let A be a 3 4 matrix, let y1 and y2 be vectors in 3, and let w = y1 + y2. Suppose y1 = Ax1 and y2... Problem 46E: Let A be a 5 3 matrix, let y be a vector in 3, and let z be a vector in 5. Suppose Ay = z. What... Problem 47E: [M] In Exercises 37-40, determine if the columns of the matrix span 4. 37. [725853496102779215] Problem 48E: [M] In Exercises 37-40, determine if the columns of the matrix span 4. 38. [574968754499911167] Problem 49E: [M] In Exercises 37-40, determine if the columns of the matrix span 4. 39.... Problem 50E: [M] In Exercises 37-40, determine if the columns of the matrix span 4. 40. [81167137856911779634187] Problem 51E Problem 52E format_list_bulleted