Linear Algebra Exam: Second Prelim - Math 2210, Exercises of Linear Algebra

INSTRUCTIONS. • This test has 6 problems on 6 pages, worth a total of. 100 points. Check if you have all 6 pages with ques-.

Typology: Exercises

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Math 2210 - Linear Algebra
Second prelim - 5 November 2013 - 7:30 to 9:00pm
Name and NetID:
Whose discussion section are you enrolled in? Circle one. Yash Lodha Wai-kit Yeung
At what time is the discussion section you enrolled in? Circle one.
1:25-2:15pm 2:30-3:20pm 3:35-4:25pm
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, clearly indicate where the remaining work is
and point out where your final answer is. You also
have a 2-sided page for scratchwork at the end.
No books or electronic devices allowed. You are al-
lowed a one-sided letter size paper of notes.
Please show all your work and justify your an-
swers.
OFFICIAL USE ONLY
1. / 15
2. / 20
3. / 15
4. / 20
5. / 10
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 exam-
ination.
Please sign below to indicate that you have read and agree to these instructions.
Signature of Student
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Math 2210 - Linear Algebra

Second prelim - 5 November 2013 - 7:30 to 9:00pm

Name and NetID:

Whose discussion section are you enrolled in? Circle one. Yash Lodha Wai-kit Yeung At what time is the discussion section you enrolled in? Circle one. 1:25-2:15pm 2:30-3:20pm 3:35-4:25pm

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, clearly indicate where the remaining work is and point out where your final answer is. You also have a 2-sided page for scratchwork at the end.
  • No books or electronic devices allowed. You are al- lowed a one-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 exam- ination. Please sign below to indicate that you have read and agree to these instructions.

Signature of Student

  1. (15 points) Let

A =

Compute a basis for each of the following spaces:

(a) The column space of A. (b) The row space of A. (c) The null space of A.

  1. (15 points) Let

A =

(a) Calculate the determinant of A. (b) Let ~r 1 , ~r 2 and ~r 3 be the 3 rows of A. What is the volume of the parallelepiped spanned by ~r 1 , 2~r 2 and 3~r 3? (c) Consider the linear transformation T (~x) = A~x. What is the volume of the paral- lelepiped spanned by T (~r 1 ), T (~r 2 ) and T (~r 3 )?

  1. (20 points) The space F of all functions R^3 → R is a vector space. Consider the subset L of linear transformations R^3 → R.

(a) Show that L is a vector subspace. (b) Compute a basis for L.

(c) Find the coordinates of T

x y z

 (^) = x + y + z in the basis from part (b).

  1. (20 points) For each of the following statements, either prove the statement is true or provide a counter example to show it is false.

(a) If A and B are similar matrices then they have the same eigenvalues.

(b) If A, P and Q are 2 × 2 matrices such that

= P −^1 AP = Q−^1 AQ, then P and Q are scalar multiples of each other. (c) There is no 3 × 3 matrix A such that both A and A − I 3 have null spaces of dimension 2. (d) Given a matrix A and a vector ~x, if A~x is an eigenvector of A then ~x is an eigenvector for A.

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