Linear Algebra Final Exam: Math 2210 - Cornell University, Exams of Algebra

The final exam for the course Math 2210 - Linear Algebra. The exam was held on May 14, 2014, in Classroom URHG01. The exam consists of 6 problems on 6 pages, worth a total of 100 points. The instructions for the exam include showing all work and justifying answers. Academic integrity is expected of all Cornell University students at all times. The exam covers topics such as eigenvalues, diagonalization, least-square line, linear transformations, and inner products.

Typology: Exams

2013/2014
On special offer
30 Points
Discount

Limited-time offer


Uploaded on 05/11/2023

sadayappan
sadayappan 🇺🇸

4.5

(15)

245 documents

1 / 9

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Math 2210 - Linear Algebra
Final Exam - 14 May 2014 - Classroom URHG01
Name and NetID:
Circle the name of your TA and your discussion session. Jeff Bergfalk Ahmad Rafiqi
1:25-2:15 2:30-3:20 3:35-4:25
INSTRUCTIONS
This test has 6 problems on 6 pages, worth a total of
100 points. Check if you have all 6 pages with ques-
tions.
If you need more space than you are given under a
question, write on the back side of the preceding sheet,
but be sure to label your work clearly and point out
where your final answer to each question is. You also
have a 2-sided page for scratchwork at the end.
No books or electronic devices allowed. You are allowed
a 2-sided letter size paper of notes.
Please show all your work and justify your an-
swers.
OFFICIAL USE ONLY
1. / 20
2. / 20
3. / 10
4. / 10
5. / 20
6. / 20
Total: / 100
Academic integrity is expected of all Cornell University students at all times, whether in
the presence or absence of members of the faculty.
Understanding this, I declare I shall not give, use, or receive unauthorized aid in this
examination.
Please sign below to indicate that you have read and agree to these instructions.
Signature of Student
pf3
pf4
pf5
pf8
pf9
Discount

On special offer

Partial preview of the text

Download Linear Algebra Final Exam: Math 2210 - Cornell University and more Exams Algebra in PDF only on Docsity!

Math 2210 - Linear Algebra

Final Exam - 14 May 2014 - Classroom URHG

Name and NetID:

Circle the name of your TA and your discussion session. Jeff Bergfalk Ahmad Rafiqi

INSTRUCTIONS

  • This test has 6 problems on 6 pages, worth a total of 100 points. Check if you have all 6 pages with ques- tions.
  • If you need more space than you are given under a question, write on the back side of the preceding sheet, but be sure to label your work clearly and point out where your final answer to each question is. You also have a 2-sided page for scratchwork at the end.
  • No books or electronic devices allowed. You are allowed a 2-sided letter size paper of notes.
  • Please show all your work and justify your an- swers.

OFFICIAL USE ONLY

Total: / 100

Academic integrity is expected of all Cornell University students at all times, whether in the presence or absence of members of the faculty. Understanding this, I declare I shall not give, use, or receive unauthorized aid in this examination. Please sign below to indicate that you have read and agree to these instructions.

Signature of Student

  1. (10+5+5 points) Consider the matrix A =

. Eigenvalues of A are 1, 1 , 4.

(1). Orthogonally diagonalize the matrix A. (2). Does there exist a unit vector x in R^3 such that xT^ Ax = 0? Does there exist a unit vector x in R^3 such that xT^ Ax = 6? If yes, find the vector. If no, explain the reason. (3). Suppose

A is also symmetric, compute

A.

Hint: Consider the orthogonal diagonalization of

A, then compute its square, and compare with the orthogonal diagonalization of A.

  1. (5+5 points) Let T : Rn^ → Rn^ be a length preserving linear transformation; that is, ‖T (x)‖ = ‖x‖ for all x ∈ Rn. (1). Show that T also preserved orthogonality; i.e., T (x)·T (y) = x·y for all x, y ∈ Rn.. Hint: consider the length square of T (x + y). (2). Show that the standard matrix of T is an orthogonal matrix, i.e., its columns form an orthonormal set.
  1. (10 points) Find the least-square line y = ax + b that best fits the following four data points (1, 0), (2, 1), (4, 2), (5, 3).
  1. (5+5+5+5 points) For each of the following statements say if it is true of false; give reasons if it is true, and a counterexample if it is false. (1). The following function defines an inner product on the vector space C^1 [0, 1],

〈f, g〉 =

0

[f (x)g(x) + f ′(x)g′(x)]dx

(2). For any two vectors x, y ∈ Rn,

‖x + y‖^2 + ‖x − y‖^2 = 2‖x‖^2 + 2‖y‖^2.

(3). Let U be an n × n orthogonal matrix, then any real eigenvalue of U must be 1. (4). If two symmetric matrices A, B ∈ Mn×n are both positive definite, then A + B is also positive definite.

This page is for scratch work; it will not be graded unless you point us here from the page where the question was posed.