Linear algebra questions
Math 220 Written Homework 1 Be sure to show all your work including each step of elimination and clearly indicate your final answer. Only use a calculator or computer algebra system on problems marked with [C]. 1. Consider each linear system below. (a) 4x + 8y − 12z = 44 3x + 6y − 8z = 32 −2x − y = −7 (b) x1 − 2×2 + x3 − x4 = 5 2×1 + x2 − 3×3 + x4 = −3 3×1 − x2 − 2×3 =0 (c) x+ y+ z =7 2x + 3y + z = 18 −x + y − 3z = 1 For the linear system: i. Find the corresponding augmented matrix. ii. Perform Gauss Jordan elimination (row reduction) on your augmented matrix. Indicate which elementary row operation was used at each step of elimination , or R1 ↔ R2. using appropriate notation such as R1 → R1 − 3R2, R3 4 iii. Give the solution set of the linear system or explain why no solution exists. 2. Suppose experimental data is represented by a set of points in the plane. An interpolating polynomial for the data is a polynomial whose graph passes through every point. (a) Set up a system of linear equations to find the interpolating polynomial y = ax2 + bx + c that passes through all three of the following data points: (1, 6), (2, 3), and (3, 2). (b) Find the augmented matrix corresponding to your linear system from part (a). (c) Solve your linear system using Gauss Jordan elimination (row reduction) and find the interpolating polynomial. 1 3. A chemist has one solution that is 20% hydrochloric acid and another solution that is 70% hydrochloric acid. How many ounces of each solution should they use to produce 20 ounces of 50% hydrochloric acid solution? 4. Consider the linear system x + hy = 2 3x + 6y = k. Determine all possible values of h and k so that the linear system has (a) no solution. (b) a unique solution. (c) infinitely many solutions. 5. The network in the figure below shows the traffic flow (in vehicles per hour) over several one-way streets. (a) Write equations that describe the traffic flow. At each intersection, set the flow in equal to the flow out. (b) Find the augmented matrix corresponding to the system of linear equations in part (a). (c) [C] (You may use a calculator or computer algebra system to do the elimination process for this problem.) Give the reduced echelon form of your augmented matrix. (d) Find the general solution of the linear system. 2 6. Suppose we want to solve the system with corresponding augmented matrix: 1 −2 b1 −2 4 b2 0 5 b2 b1 Show that the system will be inconsistent for some b = b2 in R3 . . b3 2 −1 7. Let v = and w = . Sketch v, w, v + w, and v − 2w in the Euclidean plane. 1 1 Then compute v +w and v −2w algebraically and verify your answers match your sketch. x 1 0 2 8. Show that every vector in R can be written as a linear combination of , , and y 0 1 1 in infinitely many ways.. Start your work by writing and labelling the appropriate 2 vector equation in the form x1⃗v1 + x2⃗v2 = ⃗b. 3
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