Part 1 – Linear Algebra Questions Credit will be given for: mathematically correct solutions clearly laid-out solutions to the problems posed fully worked solutions within the scope as indicated by the question. While you may of course check results to some of the questions 1-6 below using the computer – you should not include computer generated solutions for part 1. Find the magnitude of the vector s where where is a fixed constant. (2 marks) Let and be the lines with the equations where a, b and c are fixed constants. If is the angle between the lines, show that (3 marks) Find values of a, b and c such that the angle between the lines and is for all values of . (Hint: the result must be true for all values of so consider values of a, b and c which remove the dependence upon .) (2 marks) For the remainder of this question, use these values of a, b and c. The plane is given by Find the position vector of P where P is the intersection of the line and . Find also the position of Q where Q is the intersection of the line and . (4 marks) Show that as varies, the point P lies on a circle with centre Q. What is the radius of the circle? (3 marks)   Let a and b be arbitrary vectors. Prove that . (Do not use an argument involving i, j, k.) (3 marks) Verify this for (3 marks) Consider the linear map which is given by Let Find and so on. (3 marks) Do the same for . (3 marks) What do you think is happening? By considering etc. for a general , prove your result. (5 marks) Find a basis for (4 marks) Let U be the set of real 33 matrices with the following properties: the sum of all numbers in each row and in each column and in each of the two diagonals is equal to zero. Show that U is a subspace of the vector space of all 33 matrices. (5 marks)   Let be the space of polynomials of degree less than or equal to 3. Decide whether the set of polynomials forms a basis for . (4 marks) Is in the linear span of Q? (3 marks) For what value(s) of a is in the linear span of Q ? (3 marks) Your answers to the questions below must be submitted as a typed (Word) document containing (i) calculations and output from Matlab annotated with appropriate explanations/justifications (part 2) and (ii) graphs pasted from Maple, displaying full use of Word's equation editor (further criteria listed below) (part 3) and submitted to the Turnitin link which will be available in StudySpace closer to the stated deadline. Part 2 – Linear Algebra with the Aid of Matlab For part 2 you are expected to carry out the full calculations in Matlab and submit a Word document which should include computer output, suitably annotated as well as further details. Your output should show the Matlab calculations together with your explanations/justifications. Credit will be given for: Evidence of appropriate calculations performed in Matlab (copy of the commands used) Appropriate results of calculations – output from Matlab Annotation of output to highlight and explain answer presented Additional explanation/justification of the mathematical principles required to solve the problem 6. Find a subset of the following vectors which forms a basis for 5: Please indicate your reasoning clearly and carry out appropriate investigative calculations in Matlab. Please submit all relevant output with annotation – showing a clear trail of your investigation. (10 marks) 7. Many birds fly in straight lines – for example from perch to perch. Show that if a bird leaves it nest (which may be assumed to be at the origin of a 3-D Cartesian co-ordinate system) and flies only in the directions of the vectors: then it can reach any point in the locality of its nest (i.e. within its flight range and assuming that curvature of the Earth etc. may be neglected). Decide what mathematical property/result you need to demonstrate – then use Matlab to carry out the calculations. If the bird wishes to get from the nest to the point given by position vector what is the distance that it should fly along each of the 3 directions given above. Again, use Matlab to carry out the calculations to solve this problem. (7 marks) Part 3 – Mathematical Modelling with Ordinary Differential Equations Assignment Brief and Assessment Criteria (these will be discussed within a timetabled class) Credit will be given for: Accurate and logically constructed mathematical answers with sufficient detail of the steps undertaken in producing them Appropriate use of language and clear logical organisation of information Document presentation skills including use of an equation editor Inclusion of informative and appropriately labelled Maple graphs where requested in the questions 8. We saw in an early example from the differential equations lectures how to formulate the simple mathematical model for bacterial growth when this is proportional to the number of bacteria present. Here we are going to investigate whether this model (or something better) is appropriate as a model for the growth of a tumour. The notation we shall use denotes the time since the start of the tumour growth as t and number of cells in the tumour as N(t), both in suitable units dependent on specifics of the investigation. Your answers to the questions below must be as a typed document containing graphs pasted from Maple, displaying full use of Word's equation editor (further criteria listed below) and submitted to the Turnitin link which will be available in StudySpace closer to the stated deadline. (a) We begin by assuming that the growth rate is indeed proportional to the number of cells present at any time. Formulate and solve this ('by hand' in your document) for N, denoting the constant growth rate by r. State clearly any restrictions that might be appropriate for r. What does this model tell us about the size of the tumour in the long-term and therefore is this an appropriate model to use for predicting tumour sizes? (10 marks) (b) You have already encountered Maple in another module and indeed we have demonstrated in our sessions its particular use for giving analytical solutions to differential equations. Use the package to solve the ODE of part (a) as confirmation of your earlier work, then produce two solution curves on the same Maple graph for the cases r=1 and r=2 when the initial value in both cases is N=2. Plot over a sensible range for time and include a legend so that the two curves can be distinguished as well as a title before copying this into your Word solution document. (5 marks) (c) We now consider what happens if we remove the restriction that r has to be constant. In particular we want to see what happens if we allow r to reduce with time at a rate proportional to the current value of r (assume it reduces by a fraction b per unit time). (i) Formulate and solve ('by hand') the ODE for this variable r given that the initial value of r is r_0. (2 marks) (ii) Using the result you have found for r(t) in part (i), solve 'by hand' the ODE for the tumour dN/dt=r(t)N denoting the initial value as N(0)=N_0. What now happens to the number of cells in the long-term and particularly is this any improvement on the model solved in part (a)? (8 marks) (iii) Confirm your solution of (ii) by using Maple to solve the ODE as you did earlier then produce a plot of the solution. Have this ranging over sensible values of time to illustrate your observations and include the two curves for N_0=1 and N_0=2 with r_0=2 and b=1 in both cases. This plot along with the preceding Maple commands and output which solve the ODE of part (ii) must be copied into your Word document. (8 marks)