Docsity
Log inSign up
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search Store documents

The best documents sold by students who completed their studies

Search through all study resources
Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

From our blog

Exams and Study
Go to the blog

Practice Exam 2 - Linear Algebra | MATH 355, Exams of Linear Algebra

Rice UniversityLinear Algebra

Material Type: Exam; Class: LINEAR ALGEBRA; Subject: Mathematics; University: Rice University; Term: Fall 2004;

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

koofers-user-ztg
koofers-user-ztg 🇺🇸

10 documents

1 / 2

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Math 355 Exam #2 October 5, 2004
This exam consists of 2 sections. You do not have to show any work for the problems in Part
I. Solutions to problems in Part II, however, must show work to earn full credit. Pledge your
exam before turning it in. Good luck!
Part I: No justification is necessary for the following questions.
1. Let A=
1201
0013
0000
.
(a) What is the dimension of the column space of A?
(b) What is the dimension of the row space of A?
2. What is the definition of a linear transformation?
3. Let ϕ:V→Wbe a linear transformation between finite dimensional vector spaces. What
is the formula relating
•the dimension of the kernel of ϕ,
•the dimension of the image space of ϕ,
•the dimension of V.
4. Let ϕ1, ϕ2:V→Wbe linear transformations. Is the map ϕ:V→Wdefined by
ϕ(v) := ϕ1(v) + ϕ2(v) necessarily a linear transformation?
5. Let B={v1;v2;v3}be an ordered basis for a vector space V. Determine cB(v1), cB(0) and
cB(v3−v2).
6. Let v1, . . . , vkbe linearly independent vectors in a k–dimensional vector space V. Is
{v1, . . . , vk}necessarily a basis vor V?
7. Let B:= {v1, v2}be an ordered basis for Vand let ˜
B:= {v1+v2;v2}. What is the matrix
Asuch that cB(v) = Ac ˜
B(v) for all v?
Part II on reverse
pf2

Partial preview of the text

Download Practice Exam 2 - Linear Algebra | MATH 355 and more Exams Linear Algebra in PDF only on Docsity!

Math 355 Exam #2 October 5, 2004

This exam consists of 2 sections. You do not have to show any work for the problems in Part I. Solutions to problems in Part II, however, must show work to earn full credit. Pledge your exam before turning it in. Good luck!

Part I: No justification is necessary for the following questions.

  1. Let A =

(a) What is the dimension of the column space of A? (b) What is the dimension of the row space of A?

  1. What is the definition of a linear transformation?
  2. Let ϕ : V → W be a linear transformation between finite dimensional vector spaces. What is the formula relating - the dimension of the kernel of ϕ, - the dimension of the image space of ϕ, - the dimension of V.
  3. Let ϕ 1 , ϕ 2 : V → W be linear transformations. Is the map ϕ : V → W defined by ϕ(v) := ϕ 1 (v) + ϕ 2 (v) necessarily a linear transformation?
  4. Let B = {v 1 ; v 2 ; v 3 } be an ordered basis for a vector space V. Determine cB (v 1 ), cB (0) and cB (v 3 − v 2 ).
  5. Let v 1 ,... , vk be linearly independent vectors in a k–dimensional vector space V. Is {v 1 ,... , vk} necessarily a basis vor V?
  6. Let B := {v 1 , v 2 } be an ordered basis for V and let B˜ := {v 1 + v 2 ; v 2 }. What is the matrix A such that cB (v) = Ac (^) B˜ (v) for all v?

Part II on reverse

Part II:

  1. Consider v 1 :=

 (^) , v 2 :=

 (^) , v 3 :=

 (^) , v 4 :=

Among v 1 ,... , v 4 find a basis for span(v 1 ,... , v 4 ).

  1. Let V be the vector space of all polynomials of degree at most four. Determine whether f 1 := 1 − t, f 2 := 1 + 2t^2 , f 3 := t + 2t^2 are linearly independent or not.
  2. Let V be the vector space of all polynomials of degree less or equal than two. Let B be the ordered basis {1; t; t^2 }. Consider the linear transformation ϕ : V → V given by ϕ(p(t)) = p(2) · (t + 2). Which matrix represents ϕ with respect to the basis B?
  3. Let V be the vector space of all polynomials of degree less or equal than 27. (a) Show that the map ϕ : V → R given by ϕ(p(t)) := ∫^01 p(t) dt is a linear transforma- tion. (b) Show that ϕ is onto, i.e. show that the image space of ϕ is R. (c) Determine the dimension of the kernel of ϕ.