MAST10007 - Linear Algebra - Vectors - Mathematics Assignment Help

Assignment Task

Question 1

The following diagram refers to a section of a road network, involving intersections labelled A, B, C and D. The number of vehicles x1, x2, x3, x4 entering or exiting each road in a particular time period is indicated.

At each intersection, the number of vehicles going in must be equal to the number of vehicles going out. 

(a) (i) Write down a linear equation for each intersection A, B, C and D.

(ii) Given that 

find the general solution of the linear system in part (i). 

(iii) Determine the minimum number of each vehicle x1, x2, x3 and x4 in the road network. 

(b) Consider the system of equations 

2x + cy + 8z = 3 

x + 2y + cz = 1 

where c 2 R. 

(i) Explain why this system can never have a unique solution. 

(ii) Find the value of c such that there are no solutions

Question 2
(a) Let B be a m ? n matrix. Explain why BT B is always well defined.
(b) Let

Question 3
Let a = (1, 1, 1), b = (1, 1, 1) and c = (1, 1, 1).

(a) Find the vector form of the line in the direction of c that passes through the point (0, 2, 4).
(b) Find the cartesian form of the plane that passes through the point (3, 2, 1) and contains the vectors a and b.
(c) Calculate the area of the parallelogram specified by the vectors a and b.
(d) Calculate the volume of the parallelepiped formed by the vectors a, b and c.

Question 4

Question 5
(a) Let T : R2 ! R2 be a linear transformation where T(1, 0) = (1, 0) and T(0, 1) = ( 2, 1).
(i) Draw a diagram to show the e↵ect of T on the unit square with vertices (0, 0),(1, 0),(0, 1) and (1, 1). Clearly label the image of each vertex.
(ii) Determine T2(1, 0) and T2(0, 1), where T2 denotes the transformation T applied twice.
(b) Consider the linear transformation S : P2 ! P1 given by S(a0 + a1x + a2x2)=(a0 + a2) + a1x. The standard bases for P2 and P1 are {1, x, x2} and {1, x} respectively. Determine the standard matrix representation of S.

Question 6
Consider the map T : R3 ! R2 given by
T(x1, x2, x3)=(x2 - 2x3, 3x1 + x3).

(a) Show that T is a linear transformation by verifying the two defining properties.
(b) Determine the image of T.
(c) Find a basis for the kernel of T.
(d) Is T injective? Give a reason.
(e) Is T surjective? Give a reason.

Question 7
A person has three tasks to complete. Let Y denote the number of tasks that remain to be completed after X days. The following data has been collected:

X Y
1 3
2 1
4 1

(a) Using the method of least squares, find the line of best fit to the data.
(b) Draw the line of best fit on a graph, and mark in the data points.
(c) Use your answer to estimate the number of days it will take to complete the tasks.

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