MAT1102 Algebra &amp; Calculus I Tutorial &amp; Computer Laboratory: Week 9 Calculus: Finish the exercises for Week 8, and all Activities for sections now completed in lectures. Algebra: This tutorial covers Chapter 1 on Matrices. By the end of week 9 you should have mastered all the Readings &amp; Activities of Chapter 1 in your Study Book. 1. Attach solutions to the following Homework problems. Use careful language and symbols, offer explanatory statements, and conclude clearly. Larson 1.1: Q 3,7,14,27,37. Also 51,55 (Ed 4) or 55,59 (Ed 5). Larson 1.2: Q 9,13,15,17,25,27,29,31. Also do 39,41a,c,59 (Ed4) or 41,43a,c,61 (Ed 5). Larson 1.3: Q 5, 11 and 1 network problem of your choice. 2. Write the augmented matrix for the system of equations given below, and give the solutions, if any. a + 2b - c = 1 b + 3c = 3 2c = 4 Solve as above, replacing the third equation by each of the following in turn: 2c = 0, 0c = 2, 0c = 0. Write the augmented matrix in each case. 3. Two planes are defined by the following equations: 2x - y + 3z -4x + 2y - 6z = = 1 3 (1) (2) Write down the normals to these planes, and draw a rough sketch of the orientation of the planes relative to one another. Add a sketch of the possible relative position of the plane x + 2y - z = 2. Normals: Sketch: 1 Use exactly one elementary row-operation to reveal the contradiction in equations (1) and (2). Does that contradiction still arise if you reduce all 3 equations simultaneously? Explain. 4. Say how we can tell that the planes defined below are not parallel, and use a rough sketch to show how they will intersect. x + 2z y+z = = 1 3 (3) (4) Without any row-reduction, write down the parametric solution to the system above. Use it to confirm that planes (3) and (4) intersect along a line with vector equation (x, y, z) = (1 - 2t, 3 - t, t), and give a direction vector parallel to this line. Direction vector: 5. How do we know that none of the planes defined by the following equations are parallel? Use dot product to check if any are perpendicular. x+y-z 3x + 5y - z x + 5y + 3z = = = 0 10 20 (5) (6) (7) 2 Using <a href="/keyword/elementary-row-operations/" >elementary row operations</a> to reduce the augmented matrix for this system to <a href="/keyword/reduced-row-echelon/" >reduced row echelon</a> form. Work systematically from left to right, using the diagonal entry in each column to clear that column. Write out the parametric form of the solutions. In the Lab: 1. Refer to questions 3 and 5 above. Enter the augmented matrix A for the system of equations in each case, and use MATLAB to row-reduce it. Interpret the output clearly in each case. Note that if we enter a matrix A in square brackets as follows, A = [ 1 2 -1 1; 0 1 3 3; 0 0 2 4], then the command rref(A) gives its reduced row-echelon form. Also try rrefmovie(A), where you can control the pace and see each column being cleared systematically from left to right. 3) Commands: 5) Commands: Output matrix: Output matrix: Conclusion: Conclusion: Show that the planes given in Q5 intersect in a line: ie write your solution for 5) in a form that shows that it is the vector equation of a line: (x, y, z) = ( , , ) + t( , , ). Is this the same equation you found when you row-reduced by hand in Q5? 3 2. Plot the planes in Q 5 using the x and y intervals suggested below. For help with plotting surfaces with equations of form z = f (x, y), refer to p17 of your MATLAB Handbook, or use help meshgrid and help mesh. Enter values for x and y from 0 to 5. x=linspace(0,5,100); or simply x=linspace(0,5) will enter 100 such values. Enter the same or similar values for y. [X Y] = meshgrid(x,y); will prepare lists from which all possible pairs of x's and y's can be obtained. Now calculate the z-values for the...