Apm1513 applied linear algebra assignment solve the system

Question

APPLIED LINEAR ALGEBRA ASSIGNMENT

QUESTION 1 -

(a) On Matlab/Octave command line, assign values to the variables a and b, for example a = 3, and b = 5, and write statements to find the sum, difference, product, and quotient of a and b

(b) What do the following Matlab/Octave statements mean?

(i) 1 + 1 5

(ii) 1 + [1  5]

(iii) 1   5'

(iv) [1  5']

(c) Given the following vectors                 

a = [ 2    4     5 ],

b = [ 6    2    2 ],

Calculate -

(i) a.*b,

(ii) a. /b,

(iii) a. ^{^}b,

(d) Given the vector [1 2 3 4 5] Write Matlab/Octave statements that use different vector array operations to produce the following vectors

(i) [2  4  6  8  10]

(ii) [½  1  3/2  2  5/2]

(iii) [1  ½   1/3  ¼   1/5]

(iv) [1  1/2²  1/3²   1/4²  1/5²]

QUESTION 2 -

(a) Suppose we have R1000,00 saved in the bank, with interest compounded at the rate of 9% per year. Write a Matlab/Octave code to determine how much your bank balance will be after one year.

(b) Given the following system of equations

3x + 2y - z = 10

-x + 3y + 2z = 5

x - y - z  = -1

(i) Write Matlab/Octave code to solve the above system of equations.

(ii) Write the Matlab/Octave code to calculate the residual to the above system of equations.

(c) Provide definitions for the following

(i) Overdetermined system

(ii) Underdetermined system

 (iii) Conditioned system

(d) A formula for the population of the USA is

P(t) = P₀ - ae^{-0.02(t-1800)}

where t is the date in years. Some actual data is given below

Date      Population

1800       5308000

1820       9638000

1840       17069000

1870       38558000

1900       750995000

1930       122775000

1950       150697000

Write a Matlab/Octave code that finds values of P₀ and a, and that gives a best fit of the formula to the data. Then plot the graph showing the function P(t) against time t as a continuous line, together with the given data points as discrete points.

QUESTION 3 -

(a) Write an Octave/Matlab code that evaluates

₁∫³ dv/1+ v^1.8, where

v is taken to be the 0 3 after the integration

(b) Calculate the average slope of the function in (a) above in the interval a < x < b, defined

as f(b) - f(a)/b-a, where a = 1 and b = 3

(c) Use Octave to draw the following graphs

(i) f(x) = x sin (x), -3 ≤ x ≤ 8,

(ii) f (x) = cos (x²) , 0 ≤ x ≤ 5,

(d) Write an Octave/Matlab code that plots the following graph

Z = sin√(x²+y²)/√(x²+y²),

in the range -7 ≤ x ≤ 7, -11 ≤ y ≤ 11, with a contour map.

QUESTION 4 -

(a) Consider the equation

AX = B

where X^T = (x₁, x₂, . . . , x_n) and B^T = (b₁, b₂       b_n) are 1 x n vectors, and A a n x n matrix.

(i) Write Octave/Matlab commands that can be used to capture A and B into the computer memory.

(ii) Write Octave/Matlab command(s) that replaces the ith column of A with B.

(iii) Express the ratio B/A in Octave/Matlab using \ and /.

(b) Consider the system of equations                                                                                    

3x₁ + 4x₂ + 5x₃ = 4,

6x₁ + 2x₂ + 3x₃ = 7,

x₁ + 3x₂ + 3x₃ =  1

(i) Solve the system by writing an Octave/Matlab code using the Gauss-Seidal iterative method.

(ii) Find the exact, solution by Gaussian elimination i.e., by analytic means and by using either Octave or Matlab.