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MATLAB
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In Matlab, define the matrix A and vector b, such that Ax = b. Copy/paste the Matlab code and output for A and b to your document.

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

Create a word document or similar. Copy/paste the relevant code and output from the Matlab command window to the document. Please ensure each question is labelled, i.e. Question 1, Question 2, etc.

 

Once you have finished the assignment, save your document as a PDF file. Ap- pend all figures to this file and upload to iLearn.   Google ‘merge pdf’ to find online sources that merge multiple PDF files.

 

1. [5 marks] Consider the system of linear equations

 

x1 + x2 − 3x3 − 2x4 + 2x5 = 1,

−x1 + 2x3 + x4 = 1, x1 + 2x2 − x3 − x4 + 2x5 = 4,

−2x1 − 2x2 + 15x3 + 9x4 − 9x5 = −1, x1 − x2 + 2x3 − 2x4 − 3x5 = −1

for the unknown vector x = (x1, x2, x3, x4, x5)T .

(a) In Matlab, define the matrix A and vector b, such that Ax = b. Copy/paste the Matlab code and output for A and b to your document.

(b) Using the command rref determine the solution to the homogeneous system Ax = 0. Copy/paste the Matlab code and output to your document and state the solution to the homogeneous system.

(c) Using the command  A b determine the solution  to the inhomogeneous  system

Ax = b. Copy/paste the Matlab code and output to your document.

 

 

 

2. [6 marks] Consider the line given by the vector equation

 1   2 

r =  3  + t −1 for t ∈ R,

and the plane given by the Cartesian equation

 

3x − y + 5z = 7.

(a) By hand (i.e., the method presented in lectures), determine the value of t and the corresponding point (a, b, c) at which the the line intersects the plane. Add the values for t and (a, b, c) to your document.

(b) In Matlab define [x,y]=meshgrid(0:4) and using mesh(x,y,z, ‘linestyle’, ‘none’,  ‘facecolor’,  ‘red’) plot the plane. Type hold on.

(c) In Matlab define t = linspace(-2,2). Using the plot3() command, plot the line in the same figure as the plane. Add the parametric equations of the line to your document.

(d) In the same figure, plot the point of intersection (a, b, c) of the line and plane; use the command

plot3(a,b,c,‘ko’,‘markersize’,20, ‘MarkerFaceColor’,‘k’).

(e) Add an appropriate title, and x, y, z-labels to your figure and save as a PDF.

Attach the figure to the main document, using the online merge packages.

3. [4 marks] Consider the functions

f (x) = √9 − x and g(x) = x2.

(a) Determine the domain of the composite function (f ◦ g)(x). In Matlab, define the domain of f ◦ g using the linspace command, and define the composite function f ◦ g. Copy/paste the code to your document.

(b) Plot the composite function using the plot() command.

(c) Add an appropriate title, and x, y-labels to your figure and save as a PDF. Attach the figure to the main document, using the online merge packages

 

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