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quize_.pdf
Fall 2014 Math 2101-01 Quiz 1 Page 1 of 2
Due: Monday, 10/20/14 in class Name:
Time spent on this quiz: Total number of work sheets:
Note: Read the following instructions carefully before starting the test.
1. This is a take-home exam. Do NOT get help from anyone else and do NOT share your work with others. Failing to do so, you will receive NO credit.
2. Show your work and explain the reasoning, especially for crucial steps, to receive full credit.
3. Write your solution on both sides of blank papers (for the sake of trees) and initial your work sheets.
4. Arrange your work sheets in order, put the problem sheet on the top, and paper-clip them before submission.
1. (4 pts) Compute
2
1 0 −2 1 0 1
T
−3
2 −1 1 −1 1 0
2 1
−2
. Show your calcu-
lation of matrix operations.
2. A swimmer is swimming N30◦W at 2 mph in still water.
(a) (2 pts) Give the velocity of the swimmer and include a sketch. Express your answer in exact values and specify its unit.
(b) (5 pts) A current in S60◦W direction at 1 mph affects the velocity of the swimmer. Give the new velocity (in exact values) and speed (to 2 decimal places) of the swimmer and include a sketch showing how it’s related to the velocity in (a) and the current.
3. (3 pts) Use definition to determine whether vector v =
3 2 6
can be expressed
as a linear combination of vectors in S =
1 0 2
,
−1 1 2
by first writing down
a vector equation. Justify your answer.
Fall 2014 Math 2101-01 Quiz 1 Page 2 of 2
4. (3 pts) Find the vector found by rotating vector
−1√ 3
on the xy-plane counter-
clockwise for 330◦ about the origin via a matrix-vector multiplication.
5. Consider matrix A =
0 1 −3 1 −2 −1 0 1 0 2 0 2 −6 4 −6
.
(a) (5 pts) Apply the Gaussian elimination algorithm to find the reduced row echelon form R of matrix A. Specify the elementary row operation performed in each step.
(b) (2 pts) Use definition to determine the rank and nullity of matrix A.
{Remark: In case you have difficulty in the procedure of Gaussian elimination, you may use tools to find R and then continue to answer part (b).}
6. (6 pts) A commuter airline wants to purchase a fleet of 30 airplanes with a combined carrying capacity of 840 passengers. The three available types of planes carry 16, 24, and 40 passengers, respectively. Determine all possible purchase op- tions. Find how many planes of each type to be purchased if the number of 40—passengers planes purchased is the least. {Note: You should first identify the unknowns and then write a system of linear equations that leads to the answer.}
