MTH319: (a) Using matrix row operations, solve for the solution of the above system: Mathematical Methods I Assignment, SUSS, Singapore

University Singapore University of Social Science (SUSS)
Subject MTH319: Mathematical Methods I

Question 1

Given a system of linear equations as:

(a) Using matrix row operations, solve for the solution of the above system.

MTH319: (a) Using matrix row operations, solve for the solution of the above system: Mathematical Methods I Assignment, SUSS

(b) Describe the system in terms of its linear independence, consistency and rank. What does the solution represent?

(c) Hence, determine the vector equation of a line that is normal to:

MTH319: (a) Using matrix row operations, solve for the solution of the above system: Mathematical Methods I Assignment, SUSS

and contains the solution from Question 1(a).

Question 2

Figure Q2 below shows 5 data points being plotted. We want to find a best-fit line passing through these data points.

MTH319: (a) Using matrix row operations, solve for the solution of the above system: Mathematical Methods I Assignment, SUSS

Figure Q2

(a) Using linear regression by minimizing the sum of squared errors, show that the equation of the best-fit line for the above data points is:

MTH319: (a) Using matrix row operations, solve for the solution of the above system: Mathematical Methods I Assignment, SUSS

(b) Calculate the sum of squared errors for the best-fit line given in Question 2(a).

(c) Hence, determine the vector equation:

MTH319: (a) Using matrix row operations, solve for the solution of the above system: Mathematical Methods I Assignment, SUSS

which also represents the same line given in Question 2(a). Note that there is no unique answer.

Question 3

Given the matrix A below:

MTH319: (a) Using matrix row operations, solve for the solution of the above system: Mathematical Methods I Assignment, SUSS

Compute the eigendecomposition of A.

Question 4

(a) Classify the following DEs. For each DE, state its (i) type (PDE or ODE), (ii)
solution function, (iii) order, (iv) linearity and (v) homogeneity.

MTH319: (a) Using matrix row operations, solve for the solution of the above system: Mathematical Methods I Assignment, SUSS

(b) Show that the function:

MTH319: (a) Using matrix row operations, solve for the solution of the above system: Mathematical Methods I Assignment, SUSS

is a solution to the differential equation:

MTH319: (a) Using matrix row operations, solve for the solution of the above system: Mathematical Methods I Assignment, SUSS

(c) Obtain the particular solution of the IVP below, in which k is a const

MTH319: (a) Using matrix row operations, solve for the solution of the above system: Mathematical Methods I Assignment, SUSS

Determine the particular solution of the IVP below, where β, ∝ and Q are constant parameters

MTH319: (a) Using matrix row operations, solve for the solution of the above system: Mathematical Methods I Assignment, SUSS

Question 5

The populations of two species, P1(t) and P2(t), are governed by the system of ODEs: where a and b are positive constant parameters.

MTH319: (a) Using matrix row operations, solve for the solution of the above system: Mathematical Methods I Assignment, SUSS

(a) Determine the general solution of the system above.

(b) Given the information below, determine the particular solution.

MTH319: (a) Using matrix row operations, solve for the solution of the above system: Mathematical Methods I Assignment, SUSS

(c) For your solution in Question 5(b), sketch P1(t) and P2(t) for t ≥ 0 and P1, P2 ≥ 0. What can you observe about the two species?

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