1. NAME: PERSON #: LAB SECTION:
EAS 230 – Spring 2017 – Lab11 & HW11
EAS 230 – Spring 2017
Lab11 & HW11 Assignment
SCORE: /70
Directions:
Lab11 Problems: You can work on any of HW8 problems in your lab section.
HW11 Problems 𝟏 − 𝟑 are to be done at home and to be submitted in your
paper.
All problems in this assignment will be done by hand using the principles of
linear Algebra taught in class. MATLAB may be used in some parts of the
problems for comparison.
Your answers for this assignment will be submitted on paper (a “hard copy”) of
your work. You must show all of your work that lead to the final result.
Be sure to write your name, person # and LAB section on each page.
Turning-in your HW11 paper: This assignment paper must be turned-in at the
start of your lecture on (Thu or Fri, depending on your section) of the week
next to the week it was assigned on UBlearns.
2. NAME: PERSON #: LAB SECTION:
EAS 230 – Spring 2017 – Lab11 & HW11
HW11 Problems
HW11 Problem 1 (20 points)- Linear Independence
Determine whether each set of vectors (a, b, and c) below are linearly independent or
dependent. To do this, represent each set as a linear combination of column vectors 𝑐1 𝐯1 +
𝑐2 𝐯2 + ⋯ + 𝑐 𝑛 𝐯 𝑛 = 𝟎 (where 𝐯1 is the 1st vector, 𝐯2 is the 2nd vector, etc.) and solve for
𝑐1, 𝑐2 … 𝑐 𝑛.
a. 𝑆 𝑎 = {(−2, 1, 1), (3, −4, −2), (5, −10, −8)}
b. 𝑆 𝑏 = {(1, 2, 1), (1, 0, −1), (1, 1, 1)}
c. 𝑆𝑐 = {(1, 1, 2, 1), (0, 2, 1, 1), (3, 1, 2, 0)}
3. NAME: PERSON #: LAB SECTION:
EAS 230 – Spring 2017 – Lab11 & HW11
HW11 Problem 2 (30 points)-Rank, Linear Independence & Span
By hand, determine the rank of the following matrices by transforming them to row echelon
form (REF). Based on your calculations, are the row vectors linearly independent? If the rows
are linearly dependent, determine by hand a non-trivial set of values for the scalars 𝑐1 … 𝑐 𝑚
that would make 𝑐1 𝑅1 + 𝑐2 𝑅2 + ⋯ + 𝑐 𝑚 𝑅 𝑚 = 𝟎 (where 𝑅1 is the row 1 vector, 𝑅2 is the row
2 vector, etc.).
a. (4 pts) 𝐴 = [
1 2
3 4
]
b. (6 pts) 𝐵 = [
1 2 3
4 5 6
7 8 9
]
c. (4 pts) 𝐶 = [
1 2 3
4 5 6
7 8 10
]
d. (6 pts) 𝐷 = [
3 2 −1 4
1 0 2 3
−2 −2 3 −1
]
Do your work by hand and then check it in MATLAB. Show your MATLAB work.
Verify your results for matrices A, B, and C by using the determinant.
Lastly, for each matrix, plot the columns as vectors on the same set of axes. Describe the span
and linear independence of the column vectors. You may sketch your plots by hand or you
may use MATLAB or any other software to make your plots (including the software at the
links shown in the lecture notes).
4. NAME: PERSON #: LAB SECTION:
EAS 230 – Spring 2017 – Lab11 & HW11
HW11 Problem 3 (20 points)-Consistency in linear systems
Traffic Flow: In the downtown section of a certain city, two sets of one-way streets intersect
as shown in the figure below. The average hourly volume of traffic entering and leaving this
section during rush hour is known and is given in the diagram. The amount of traffic between
each of the four intersections is unknown (𝑥1, 𝑥2, 𝑥3 and 𝑥4).
a. (6 pts) Determine the linear system of equations in terms of 𝑥1, 𝑥2, 𝑥3 and 𝑥4. (Hint:
At each intersection, the number of automobiles entering must be the same as the
number leaving.)
b. (2 pt) Rewrite the system of equations in 𝐴𝐱 = 𝐛 form.
c. (6 pts) By hand, find rank(𝐴) and rank(𝐴|𝐛) by using Gaussian elimination to reduce
𝐴 and (𝐴|𝐛) to row echelon form (REF). Determine whether the system of equations
is consistent or inconsistent.
d. (6 pts) If the system is consistent, find the solution (i.e. determine 𝑥1, 𝑥2, 𝑥3 and 𝑥4)
by performing back-substitution.
