Question 1

Given a system of linear equations as:

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

(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:

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.

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:

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

(c) Hence, determine the vector equation:

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:

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.

(b) Show that the function:

is a solution to the differential equation:

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

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

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.

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

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

(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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