Repeat Exercise 5 for a nonzero 3 × 2 matrix.Exercise

Problem 34E [M] In a wind tunnel experiment, the force on a projectile due to air resistance was measured at different velocities: Velocity (100 ft/sec) 0 2 4 6 8 10 Force (100 lb) 0 2.90 14.8 39.6 74.3 119 Find an interpolating polynomial for these data and estimate the force on the projectile when the projectile is traveling at 750 ft/sec. Use p(t) = a0 + a1t + a2t2 + a3t3 + a4t4 + a5t5. What happens if you try to use a polynomial of degree less than 5? (Try a cubic polynomial, for instance.)5

Problem 1E In Exercises 1 and 2, determine which matrices are in reduced echelon form and which others are only in echelon form.

Problem 3E Row reduce the matrices in Exercises 3 and 4 to reduced echelon form. Circle the pivot positions in the final matrix and in the original matrix, and list the pivot columns.

Problem 2E In Exercises 1 and 2, determine which matrices are in reduced echelon form and which others are only in echelon form.

Problem 4E Row reduce the matrices in Exercises 3 and 4 to reduced echelon form. Circle the pivot positions in the final matrix and in the original matrix, and list the pivot columns.

Problem 5E Describe the possible echelon forms of a nonzero 2 × 2 matrix. Use the symbols ?, * and 0, as in the first part of Example 1.

Problem 6E Repeat Exercise 5 for a nonzero 3 × 2 matrix. Exercise 5: Describe the possible echelon forms of a nonzero 2 × 2 matrix. Use the symbols ?, * and 0, as in the first part of Example 1.

Problem 7E Find the general solutions of the systems whose augmented matrices are given in Exercises 7–14.

Problem 8E Find the general solutions of the systems whose augmented matrices are given in Exercises 7–14.

Problem 9E Find the general solutions of the systems whose augmented matrices are given in Exercises 7–14.

Problem 11E Find the general solutions of the systems whose augmented matrices are given in Exercises 7–14.

Problem 10E Find the general solutions of the systems whose augmented matrices are given in Exercises 7–14.

Problem 13E Find the general solutions of the systems whose augmented matrices are given in Exercises 7–14.

Problem 12E Find the general solutions of the systems whose augmented matrices are given in Exercises 7–14.

Problem 14E Find the general solutions of the systems whose augmented matrices are given in Exercises 7–14.

Problem 15E Exercises 15 and 16 use the notation of Example 1 for matrices in echelon form. Suppose each matrix represents the augmented matrix for a system of linear equations. In each case, determine if the system is consistent. If the system is consistent, determine if the solution is unique.

Problem 16E Exercises 15 and 16 use the notation of Example 1 for matrices in echelon form. Suppose each matrix represents the augmented matrix for a system of linear equations. In each case, determine if the system is consistent. If the system is consistent, determine if the solution is unique.

Problem 17E In Exercises 17 and 18, determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.

Problem 18E In Exercises 17 and 18, determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.

Problem 19E In Exercises 19 and 20, choose h and k such that the system has (a) no solution, (b) a unique solution, and (c) many solutions. Give separate answers for each part. x1 + hx2 = 2 4x1 + 8x2 = k

Problem 21E a. In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations. b. The row reduction algorithm applies only to augmented matrices for a linear system. c. A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix. d. Finding a parametric description of the solution set of a linear system is the same as solving the system. e. If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated linear system is inconsistent.

Problem 23E Suppose the coefficient matrix of a linear system of four equations in four variables has a pivot in each column. Explain why the system has a unique solution.

Problem 22E a. The reduced echelon form of a matrix is unique. b. If every column of an augmented matrix contains a pivot, then the corresponding system is consistent. c. The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process. d. A general solution of a system is an explicit description of all solutions of the system. e. Whenever a system has free variables, the solution set contains many solutions.

Problem 24E Suppose a system of linear equations has a 3× 5 augmented matrix whose fifth column is not a pivot column. Is the system consistent? Why (or why not)?

Problem 25E Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.

Problem 20E In Exercises 19 and 20, choose h and k such that the system has (a) no solution, (b) a unique solution, and (c) many solutions. Give separate answers for each part. x1 – 3x2 = 1 2x1 + hx2 = k

Problem 27E Restate the last sentence in Theorem 2 using the concept of pivot columns: “If a linear system is consistent, then the solution is unique if and only if ____________________.”

Problem 28E What would you have to know about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution?

Problem 26E Suppose a 3 × 5 coefficient matrix for a system has three pivot columns. Is the system consistent? Why or why not?

Problem 29E A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system. Can such a system have a unique solution? Explain.

Problem 30E Give an example of an inconsistent underdetermined system of two equations in three unknowns.

Problem 31E A system of linear equations with more equations than unknowns is sometimes called an overdetermined system. Can such a system be consistent? Illustrate your answer with a specific system of three equations in two unknowns.

Problem 33E Suppose experimental data are represented by a set of points in the plane. An interpolating polynomial for the data is a polynomial whose graph passes through every point. In scientific work, such a polynomial can be used, for example, to estimate values between the known data points. Another use is to create curves for graphical images on a computer screen. One method for finding an interpolating polynomial is to solve a system of linear equations. Find the interpolating polynomial p(t) = a0 + a1t + a2t2 for the data (1, 6), (2, 15), (3, 28). That is, find a0, a1, and a2 such that a 0 + a1(1) + a2(1)2 = 6 a 0 + a1(2) + a2(2)2 = 15 a 0 + a1(3) + a2(3)2 = 28

Problem 32E Suppose an n × (n + 1) matrix is row reduced to reduced echelon form. Approximately what fraction of the total number of operations (flops) is involved in the backward phase of the reduction when n = 20? when n = 200?