25. Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent. 26. Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot in each column. Explain why the system has a unique solution. 27. 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 28. 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? mulo 29. A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system. Suppose that such a system happens to be consistent. Explain why there must be an infinite number of solutions. 30. Give an example of an inconsistent underdetermined system of two equations in three unknowns. 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. WEB 33. Find the interpolating polynomial p(t) = ao+at+ a₂t² for the data (1, 12), (2, 15), (3, 16). That is, find ao, a₁, and a₂ such that de ao + a₁(1) + a₂(1)² = 12 mnoh 31. A system of linear equations with more equations than un- knowns 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. may *1 32. Suppose an nx (n+1) matrix is row reduced to reduced echelon form. Approximately what fraction of the total num- ber of operations (flops) is involved in the backward phase of the reduction when n = 30? when n = 300?dowell Suppose experimental data are represented by a set of points. in the plane. An interpolating polynomial for the data is a The general solution of the system of equations is the line of intersection of the two planes. 1.2 Row Reduction and Echelon Forms 23 ao + a₁ (2) + a₂(2)² = 15 ao + a₁ (3) + a₂ (3)² = 16 10 mustava 34. [M] In a wind tunnel experiment, the force on a projectile due to air resistance was measured at different velocities: 0 2 4 6 8 10 tower Velocity (100 ft/sec) Force (100 lb) 0 2.90 14.8 39.6 74.3 119 Au Find an interpolating polynomial for these data and estimate the force on the projectile when the projectile is travel- ing at 750 ft/sec. Use p(t) = ao+at+azt² + a3t³ + atª + ast5. What happens if you try to use a polynomial of degree less than 5? (Try a cubic polynomial, for instance.)5 HORDO 0-8 1-1 5 Exercises marked with the symbol [M] are designed to be worked with the aid of a "Matrix program" (a computer program, such as MATLAB, Maple, Mathematica, MathCad, or Derive, or a programmable calculator with matrix capabilities, such as those manufactured by Texas Instruments or Hewlett-Packard). gadgge SOLUTIONS TO PRACTICE PROBLEMS 1. The reduced echelon form of the augmented matrix and the corresponding system are T 3) and mish it from X1 X2 The basic variables are x₁ and x2, and the general solution is x₁ = -3 + 8x3 x₂ = -1 + x3 x3 is free d -8.x3 = -3 X3 = -1 andey lees tad tilh Note: It is essential that the general solution describe each variable, with any param- eters clearly identified. The following statement does not describe the solution: