Problem 1. In this question we use the vectors of R5.
u = (1, 0, 1, 1, 1), v = (3, 2, 1, 1, 1) and w = (0, 1, 1, 1, 2).
(a) Calculate u + v,
(b) Calculate 5u,
(c) Calculate 2u 3v,
(d) Calculate u v,
(e) Calculate kuk,
(f) Calculate kvk,
(g) Calculate the angle between u and v,
(h) Show that w is a linear combination of u and v,
(i) Find a vector in R5 which is not a linear combination of u and v. Verify that your vector is not a linear combination of u and v.
Problem 2. In this question we consider the following six points in R3.
A(0, 10, 3),B( 4, 18, 5),C(1, 1, 1),D( 1, 0, 1),E(0, 1, 3) and F(2, 6, 2).
(a) Find a vector equation for the line through the points A and B,
(b) Find general equations for the line from (a),
(c) Find a vector equation for the plane through the points C, D and E,
(d) Find a general equation for the plane from (c),
(e) Find the point where the line from (a) intersects the plane from (c),
(f) Explain why the line from (a) is orthogonal to the plane from (c),
(g) Show that the point F lies on the line from (a),
(h) Determine the shortest distance from the point F to the plane from (c).
Problem 3. Consider the following system of simultaneous linear equations.
w + x + y + z = 5
w + 2x + y + 2z = 10
w + x + y z = 3
3w + 4x 5y 5z = 1
(a) Set up an augmented matrix which represents this system.
(b) Transform your augmented matrix to row echelon form and hence nd the solutions of the system of linear equations.
(c) The solution of your equations represents an object in R4. What is this object?
Problem 4. Consider the following system of simultaneous linear equations.
w + x + y + z + v = 5
w + 2x + y + 2z + 2v = 10
w + x + y z v = 3
3w + 4x 5y 5z 5v = 1
Compare this system of linear equations with the system in exercise 3.
(a) Without setting up the augmented matrix in this case use your answer to exercise 3 to find the general solution of the system in this exercise. You should use your working from exercise 3.
(b) What is the geometric nature of the geometric object which represents the solution of this system in R5?
Problem 5. In the course MATHS108 there are six assessment components contributing to the nal grade.
- 3 assignments each given a mark out of 10 and counting a total of 9% toward the fi nal grade,
- 10 tutorials each given a mark out of 1 and counting a total of 5% toward the final grade,
- 4 quizzes each given a mark out of 6 and counting a total of 4% toward the final grade,
- 2 Matlab worksheets each given a mark out of 1 and counting a total of 2% toward the final grade,
a test given a mark out of 20 and counting 20% toward the nal grade, and an exam given a mark out of 40 and counting 60% toward the nal grade.
Suppose that a student scores a marks (out of 30) for assignments, t marks (out of 10) for tutorials, q marks (out of 24) for quizzes, m marks (out of 2) for Matlab worksheets, s marks (out of 20) for the test and e marks (out of 40) for the nal exam.
(a) Let u 2 R6 be the vector (a, t, q,m, s, e). Write down a second vector v 2 R6 so that the components of v give a weighting vector chosen so that the student's total assessment mark as a percentage is the scalar product u v.
(b) What is the scalar product u v?
(c) Student A notices that he has scores in the rst ve categories of 16, 8, 4, 0 and 11 respectively. What mark does he need to obtain in the nal exam in order to score a total of 70% for the entire course?
(d) Student B notices that she has scores in the rst ve categories of 15, 7, 10, 2 and 14 respectively. Is it possible for her to score a total of 85% for the entire course? Explain your answer.