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Column Space - Linear Algebra and Multivariable Calculus - First Midterm Exam, Exams of Calculus

This is the First Midterm Exam of Linear Algebra and Multivariable Calculus which includes System of Equations, Three Vectors, Collection, Reduced Echelon Form etc. Key important points are: Column Space, Basis, Null Space, Basis, Solutions, Equation, Linearly Independent Vectors, Set of Vectors, Reduced Row, Echelon Form

Typology: Exams

2012/2013

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Math 51 - Autumn 2011 - Midterm Exam I
Name:
Student ID:
Select your section:
Jonathan Campbell Elizabeth Goodman Julio Gutierrez
ACE (1:15–3:05pm) 03 (11:00–11:50am) 15 (11:00–11:50am)
17 (1:15–2:05pm) 25 (1:15–2:05pm)
Seungki Kim Kenji Kozai Yuncheng Lin
08 (10:00–10:50am) 14 (10:00–10:50am) 02 (11:00–11:50am)
09 (11:00–11:50am) 24 (1:15–2:05pm) 05 (1:15–2:05pm)
Michael Lipnowski Jeremy Miller Ho Chung Siu
21 (11:00–11:50am) 11 (1:15–2:05pm) 20 (10:00–10:50am)
23 (1:15–3:05pm) 18 (2:15–3:05pm) 06 (1:15–2:05pm)
Signature:
Instructions:
•Print your name and student ID number, select your section number and TA’s
name, and write your signature to indicate that you accept the Honor Code.
•There are 9 problems on the pages numbered from 1 to 11, for a total of 100
points. Point values are given in parentheses. Please check that the version of
the exam you have is complete, and correctly stapled.
•Read each question carefully. In order to receive full credit, please show all of
your work and justify your answers unless you are explicitly instructed not to.
•You do not need to simplify your answers unless specifically instructed to do
so. You may use any result from class that you like, but if you cite a theorem
be sure to verify the hypotheses are satisfied.
•You have 90 minutes. This is a closed-book, closed-notes exam. No calcu-
lators or other electronic aids will be permitted. If you finish early, you must
hand your exam paper to a member of teaching staff.
•If you need extra room, use the back sides of each page. If you must use extra
paper, make sure to write your name on it and attach it to this exam. Do not
unstaple or detach pages from this exam.
•It is your responsibility to arrange to pick up your graded exam paper from your
section leader in a timely manner. You have only until October 28, 2011, to
resubmit your exam for any regrade considerations; consult your section leader
about the exact details of the submission process.
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Math 51 - Autumn 2011 - Midterm Exam I

Name:

Student ID:

Select your section: Jonathan Campbell Elizabeth Goodman Julio Gutierrez ACE (1:15–3:05pm) 03 (11:00–11:50am) 15 (11:00–11:50am) 17 (1:15–2:05pm) 25 (1:15–2:05pm) Seungki Kim Kenji Kozai Yuncheng Lin 08 (10:00–10:50am) 14 (10:00–10:50am) 02 (11:00–11:50am) 09 (11:00–11:50am) 24 (1:15–2:05pm) 05 (1:15–2:05pm) Michael Lipnowski Jeremy Miller Ho Chung Siu 21 (11:00–11:50am) 11 (1:15–2:05pm) 20 (10:00–10:50am) 23 (1:15–3:05pm) 18 (2:15–3:05pm) 06 (1:15–2:05pm)

Signature:

Instructions:

  • Print your name and student ID number, select your section number and TA’s name, and write your signature to indicate that you accept the Honor Code.
  • There are 9 problems on the pages numbered from 1 to 11, for a total of 100 points. Point values are given in parentheses. Please check that the version of the exam you have is complete, and correctly stapled.
  • Read each question carefully. In order to receive full credit, please show all of your work and justify your answers unless you are explicitly instructed not to.
  • You do not need to simplify your answers unless specifically instructed to do so. You may use any result from class that you like, but if you cite a theorem be sure to verify the hypotheses are satisfied.
  • You have 90 minutes. This is a closed-book, closed-notes exam. No calcu- lators or other electronic aids will be permitted. If you finish early, you must hand your exam paper to a member of teaching staff.
  • If you need extra room, use the back sides of each page. If you must use extra paper, make sure to write your name on it and attach it to this exam. Do not unstaple or detach pages from this exam.
  • It is your responsibility to arrange to pick up your graded exam paper from your section leader in a timely manner. You have only until October 28, 2011, to resubmit your exam for any regrade considerations; consult your section leader about the exact details of the submission process.

Problem 1. (10 pts.) Let

A =

[

]

a) Find a basis for the column space of A.

b) Find a basis for the null space of A.

c) Write down all solutions x (if any) of the equation Ax = b for

i) b =

[

]

; ii) b =

[

]

Problem 3. (10 pts.) Consider the following matrix A and its reduced row echelon form rref(A):

A =

rref(A) =

(You do not need to check that the row reduction is correct.)

a) Find a basis for N (A).

b) Find a basis for C(A).

Problem 4. (15 pts.) Consider the matrix

A =

a 6 2 0 9 5

a) For what value(s) of a will a row interchange be necessary during row reduction?

b) Is there a value of a for which N (A) is nontrivial? Justify your answer.

c) Calculate the rank of A for each value of a (your answer may depend on the particular value of a, of course).

Problem 6. (10 pts.) Let V be the two-dimensional plane in R^4 spanned by the vectors

   

 and

Let W be the set of all vectors w ∈ R^4 such that v · w = 0 for all v ∈ V.

a) Show that W is a subspace of R^4.

b) Find a basis for W.

Problem 8. (10 pts.) Assume that m < n. Give a clear and accurate explanation why any homogeneous system of m linear equations in n unknown variables of the form Ax = 0, where A is an m-by-n matrix, always has solutions x which are not equal to the 0 vector.

The following boxes are strictly for grading purposes. Please do not mark.

Question Score Maximum

Total 100