You are time-travelers. You will almost surely be working in the year 2050. Things will change dramatically. To have any chance of significant success in life, you will need to learn a great deal and you will probably be required to reinvent yourself many times. You should start now! Develop good habits.
Math 250 is a straightforward course. One aim of the exam was to minimize arithmetic computations. Machines will likely do such tasks and therefore you need to be able to "think". There were some computational questions but they were minimal. Most of the exam asked you about the logic of the subject and, a few times, to explain your conclusions. Currently machines can't do such things well. There are many, many students worldwide who want do these things very well, and you will be competing with those students for jobs. If you want meaningful success, especially in this competition, you must practice!
About a third of the students are absent on any day of the course. And about a third of the students who attend spend much of their class time dreaming, texting, and not concentrating: yes, it is that obvious. Attending a math lecture is fairly hard work, even for a relatively straightforward topic. If you want to learn, you must participate actively, ask questions when you don't understand, read the textbook before and after lectures, and do homework. You can consult me and work with other students. I have made these recommendations several times.
Grading in this course will be simple – this is a low-level math course with no way to earn any "extra" credit. Magic opportunities to improve grades will not appear near the end of the semester. A colleague of mine remarked that I don't give grades but I do attempt to report student performance accurately. You give yourself grades. Students who are listed as (possibly!) graduating seniors should note this especially.
Problem | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 | #9 | #10 | #11 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Max grade | 16 | 10 | 8 | 10 | 8 | 4 | 12 | 10 | 8 | 6 | 8 | 99 | Min grade | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 20 | Mean grade | 7 | 7.05 | 7.17 | 4.12 | 2.53 | 3.04 | 6.93 | 3.2 | 4.73 | 2.19 | 4.76 | 52.73 | Median grade | 7 | 8 | 8 | 4 | 2 | 4 | 6 | 2 | 5 | 2 | 8 | 52 |
The versions of the exam were very similar.
Numerical grades will be retained for use in computing the final letter grade in the
course.
Here are approximate letter grade assignments for this exam:
Letter equivalent | A | B+ | B | C+ | C | D | F |
---|---|---|---|---|---|---|---|
Range | [85,100] | [80,84] | [70,79] | [65,69] | [55,64] | [50,54] | [0,49] |
Grading guidelines
The grader will read only what is written and not attempt to
guess or read the mind of the student.
The student should solve the problem given and should not invent
another problem and request credit for working on that problem.
Problem 1 (16 points)
2 points for specifying the Animal.
a) (2 points) For the answers.
b) (4 points) For the answers.
c) (4 points) For the span.
d) (4 points) For the span.
Problem 2 (10 points)
2 points for specifying the Animal.
2 points for the answer (linearly dependent). 2 points for a
non-trivial linear combination. 4 points for supporting work.
Problem 3 (8 points)
2 points for the answer (the rank and nullity of the student's
result!), and 6 points for supporting work (row operations).
Problem 4 (8 points)
a) 2 points for a correct answer, and 3 points for supporting
evidence.
b) 5 points for an explanation.
Problem 5 (8 points)
a) (4 points) 1 point for taking the transpose of B–1
and 3 points for the correct multiplication. –2 for an error in
the order.
b) (4 points) 2 points for multiplying the given column vector by
B–1, and 2 points for multiplying the result by
A–1. –2 for an error in the order.
Problem 6 (4 points)
1 point for a 5x4 matrix, M. The rank of M is worth 1 point, the
nullity of M is worth 1 point, and the nullity of MT is
worth 1 point.
Problem 7 (8 points)
a) (9 points) Writing [W|I4] is worth 2 points. Row
operations are worth 5 points. The correct answer is worth 2 points.
b) (3 points) Writing W multiplied by W–1 and getting
I4.
Ethical comment A number of students had an incorrect candidate
for W–1 yet reported that the product of that matrix
with W is I4. This certainly can't be a correct or accurate
computation. I note that individuals who are credentialed as
professional engineers are supposed to have ethical standards:
would you want to work with or for someone who reported a false
product of two matrices because it was convenient? ... or drive over a bridge designed by such a person or use a
heart stent built by ... ETC.
Problem 8 (10 points)
2 points for specifying the Animal.
2 points for clearly indicating which w's will imply that the span is
all of R4. 4 points for describing why the "good" w's do
indeed give all of R4 and 2 points for indicating why the
"bad" w does not give all of R4.
Problem 9 (8 points)
a) (4 points) 1 point for the answer, and 3 points for justification
(an explanation).
b) (4 points) 1 point for the answer, and 3 points for justification
(an example).
Problem 10 (6 points)
a) (2 points) The definition.
b) (4 points) 2 points for a valid example, and 2 points for evidence
supporting the assertion.
Problem 11 (8 points)
The explanation.
Problem | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 | Total |
---|---|---|---|---|---|---|---|---|---|
Max grade | 15 | 10 | 15 | 6 | 12 | 12 | 12 | 18 | 100 | Min grade | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 19 | Mean grade | 10.95 | 5.64 | 11.95 | 2.71 | 6.5 | 8.24 | 7.20 | 12.77 | 65.91 | Median grade | 11 | 6 | 14 | 2 | 6 | 8.5 | 8.5 | 14.5 | 70.5 |
Numerical grades will be retained for use in computing the final letter grade in the
course.
Here are approximate letter grade assignments for this exam:
Letter equivalent | A | B+ | B | C+ | C | D | F |
---|---|---|---|---|---|---|---|
Range | [85,100] | [80,84] | [70,79] | [65,69] | [55,64] | [50,54] | [0,49] |
Problem 1 was given for essentially the third time to students in these sections. Problem 4 was given previously on a quiz. Problem 5 was given in review material, as were parts a) and b) of problem 6 while part c) was given previously on the first exam. Problem 8 was given previously on a quiz. That's 63 points out of 100 on this exam!
The other problems generally resembled examples discussed in class and the textbook and/or given for homework.
Grading guidelines
The grader will read only what is written and not attempt to
guess or read the mind of the student.
The student should solve the problem given and not invent
another problem and request credit for working on that problem.
Problem 1 (15 points)
5 points for each part: 1 point for the dimension, and 4 points for
the basis requested (2 for each vector).
In part c), 1 point will
be deducted for reporting row vectors, since "vector" in this course
is supposed to refer to column vectors.
Problem 2 (10 points)
5 points for the definition, which should be given in complete English
sentences. 5 points for showing exactly how the definition fails for
the collection of vectors described.
Problem 3 (15 points)
a) The characteristic polynomial earns 2 points, and each eigenvalue
earns 1 point.
b) Each eigenvector earns 2 points.
c) 1 point for each of D and P.
d) 3 points for P–1.
e) 1 point for PD and 1 point for successfully computing
PDP–1 and getting A.
Problem 4 (6 points)
2 points for the answer and 4 points for an explanation.
Problem 5 (12 points)
6 points for each part. 1 of those is for the answer, and 5 for the
process.
Problem 6 (12 points)
In each part, 1 point for a correct answer and 3 points for some
adequate supporting explanation.
Problem 7 (12 points)
Indicating an intention to compute det(A–μB) with correct
entries in the 3x3 matrix earns 3 points. Computing the determinant
correctly earns 5 points. Getting the roots and stating a correct
answer to the problem is worth the last 4 points.
Problem 8 (18 points)
In each part, 1 point for the answer (YES/NO), 1 point for the
eigenvalue, and 4 points for some adequate supporting explanation.
The final exam
The exam seems to have been more difficult than intended. In
particular, questions 6 and 17 did not get good scores and probably
were new to people. These questions do resemble some real uses of
linear algebra. In recognition of the difficulty, I've decided to rescale the totals proportionately, so that 180
would become 200 (that is, I modified the total by multiplying that
score by 2/1.8). Using those numbers, statistical results for the
total were: high=172.22, low=55.56, mean=128.93, and
median=134.44. The raw score results (not rescaled) were
high=155, low=50, mean=116.03, and median=121.
The rescaled numerical grades will be used in computing the final letter grade in the
course.
Here are approximate letter grade assignments for this exam:
Letter equivalent | A | B+ | B | C+ | C | D | F |
---|---|---|---|---|---|---|---|
Range | [170,200] | [160,169] | [140,159] | [130,139] | [110,129] | [100,109] | [0,99] |
Grading guidelines
The grader will read only what is written and not attempt to guess
what the student's solution is or read the mind of the student.
The student should solve the problem given and should not invent
another problem and request credit for working on that problem.
Problem 1 (12 points)
a) 4 points for the definition. Only 2 points are earned if the phrase
"linear combination" is used with no explanation. These 2 points can
be recouped if the student defines "linear combination" correctly in
part a) of problem 15.
b) 8 points. 2 points of these are awarded for explaining what the
matrix manipulations have to do with the requested verification.
Problem 2 (18 points)
9 points for each of a) and b): 2 points for the dimension, 2 points
for one vector in a basis, and 5 points for the other (orthogonality
earns 3 of those points).
Problem 3 (16 points)
a) 4 points: 2 for the characteristic polynomial, and each eigenvalue
earns 1 point.
b) 5 points: each eigenvector earns 2 points, and checking
orthogonality successfully earns 1 point.
c) 4 points: 1 point for D and 1 for a possible P, and 1 point each
for correct normalization.
d) 1 point for P–1.
e) 2 points: 1 for PD and 1 for multiplying the result by
P–1.
Problem 4 (6 points)
6 points for an explanation using results and language of the course,
and related to the specific situation of this problem. A simple quote
of a textbook or lecture result will earn only 2 points.
Problem 5 (8 points)
1 point for the value of the determinant, and 7 points for the process
(good luck if you expand along a row or column!).
Problem 6 (18 points)
Obtaining the (general) characteristic polynomial is worth 2
points. Getting eigenvalues using the quadratic formula earns 4
points. There are three cases. Analysis of each case is worth 4
points: real distinct roots, complex roots with non-zero imaginary
part (square root of a negative number), and double root (multiplicity
2).
Problem 7 (10 points)
a) 5 points: 1 point for the answer, and 4 points for justification
(an explanation).
b) 5 points: 1 point for the answer, and 4 points for justification
(an example).
Problem 8 (10 points)
a) 5 points: 1 point for taking the transpose of B–1,
1 point for reversing the order, and 3 points for the correct
multiplication. If the multiplication is correct but the order is
wrong, 3 points are earned.
b) 5 points: 1 point for reversing the order, 2 points for multiplying
the given column vector by B–1, and 2 points for
multiplying the result by A–1. If the multiplications
are correct but the order is wrong, 3 points are earned.
Problem 9 (12 points)
a) 8 points: 6 points for v1 (an answer needs to be
"instantiated", with all numbers, but an explicit answer does not need
to be evaluated) and 2 points for v2.
b) 2 points for the answer, which need not be simplified.
c) 2 points for the answer.
Problem 10 (10 points)
Rewriting things as dot products earns 3 points, and then the balance
is obtained by completing the explanation. 1 point off if no ·
is used (the "dot" in dot product), and 2 points off if || ||'s
are misused.
Problem 11 (12 points)
a) 7 points. Note that the answer can be checked! 2 points for
the answer alone with no method indicated.
b) 2 points for the answer (this can use the student's
A–1).
c) 3 points: 1 point for the answer, and 2 for some explanation.
Problem 12 (12 points)
In each part, 6 points: 1 point for the answer (YES/NO) and 1 point for the
associated eigenvalue. 4 points for some explanation.
Problem 13 (10 points)
a) 6 points: 2 points for Av and 4 points for ||Av|| (of those, 2
points are earned for asserting or using
v1·v2=0).
b) 4 points for the answer.
Problem 14 (10 points)
2 points for the rank, 2 points for the nullity, and 6 points for the
process.
Problem 15 (12 points)
a) 4 points for the definition.
b) 8 points: 2 points for the answer, and 6 points for the process.
Problem 16 (12 points)
a) 7 points: AB computed is worth 5 points, and an appropriate
comment about BA is worth 2 points.
b) 5 points: knowing that det=0 is connected to non-invertibility is
worth 2 points, and then using this along with knowledge about how det
works with matrix product is worth the other 3 points. The remark that
a product of an invertible matrix with a non-invertible matrix is not
invertible earns 2 points. No points are earned if some formula such as
(AB)–1=B–1A–1 was
quoted.
Problem 17 (12 points)
a) 5 points: computation of AP and PA is worth 3 points. The
conclusion correctly derived is worth 2 points.
b) 5 points: computation of AQ and QA is worth 3 points. The
conclusion correctly derived is worth 2 points.
c) 2 points for remarks supporting the conclusion.
Course grades
I added the two in-class exam scores and the rescaled final exam scores. To that I added 50/90
multiplied by the total quiz score for each student, and then I added
50/212 multiplied by the total homework score for each student. The
total possible score which could be obtained was 500 points. I then
put these total scores in bins which corresponded to the bins used for
each of the exams (so the boundaries would be {250|275|325|350|400|425}
for F/D/C/C+/B/B+/A).
Rutgers regulations require that the exams be kept for a year. Students may look at their exams and check the grading.
Maintained by greenfie@math.rutgers.edu and last modified 5/11/2011.