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Linear Algebra Exam 1 Review for Math 333 - Fall 2007, Exams of Linear Algebra

A review for exam 1 of the linear algebra course (math 333) offered in the fall 2007 semester. The review covers sections 1.1-1.6 from the textbook, definitions, theorems, computational exercises, and proof techniques. Students are expected to understand the concepts and be able to apply them to various problems.

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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Math 333 - Linear Algebra
Exam 1 Review
Fall 2007
1 Sections Covered
The exam will cover Sections 1.1-1.6 from the textbook. You should read and understand
all of these sections.
2 Definitions
You should know ALL definitions covered in the above mentioned sections. Pay special
attention to the definitions of each of the following terms. DEFINITELY KNOW (i.e.
MEMORIZE) THESE DEFINITIONS.... Also, if asked to give a definition of something,
you can NOT give a theorem. For example, the definition of a subspace is NOT a subset
of a vector space which is closed under the operations and contains the zero vector. This
is a theorem, not the definition.
Chapter 1: Subspaces, Linear combination, Linear independence, V= span(S)” or
Sgenerates V”, Basis, Dimension
3 Theorems
You should know the statements and understand the usefulness of all theorems in the above
mentioned sections. Pay special attention to each of the following theorems. You will NOT
be asked to write the precise statements of most of these theorems, so don’t attempt to
memorize exactly what they say. Simply know how to use them. However, you may be
asked to write the precise statement if it has a ** next to it.
Chapter 1: Theorems 1.3**, 1.8, 1.9, Cor. 1 and 2 (pg. 46-47), 1.11
4 Computations and other Exercises
You are responsible for knowing any homework exercise that has been assigned as well as
similar computational exercises. However, you may choose to focus on the following types.
1. Determine if a set Vis a vector space under specific operations.
2. Determine if a subset Wof a vector space Vis a subspace of V.
3. Determine if a subset Sof a vector space Vis a basis of V, and determine if a vector
vspan(S).
4. Be able to do the following exercises from the textbook: Sec. 1.2: (9, 11), Sec. 1.4:
(13, 15), Sec. 1.5: (12), Sec. 1.6: (12).
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Math 333 - Linear Algebra

Exam 1 Review

Fall 2007

1 Sections Covered

The exam will cover Sections 1.1-1.6 from the textbook. You should read and understand all of these sections.

2 Definitions

You should know ALL definitions covered in the above mentioned sections. Pay special attention to the definitions of each of the following terms. DEFINITELY KNOW (i.e. MEMORIZE) THESE DEFINITIONS.... Also, if asked to give a definition of something, you can NOT give a theorem. For example, the definition of a subspace is NOT a subset of a vector space which is closed under the operations and contains the zero vector. This is a theorem, not the definition. Chapter 1: Subspaces, Linear combination, Linear independence, “V = span(S)” or “S generates V ”, Basis, Dimension

3 Theorems

You should know the statements and understand the usefulness of all theorems in the above mentioned sections. Pay special attention to each of the following theorems. You will NOT be asked to write the precise statements of most of these theorems, so don’t attempt to memorize exactly what they say. Simply know how to use them. However, you may be asked to write the precise statement if it has a ** next to it. Chapter 1: Theorems 1.3**, 1.8, 1.9, Cor. 1 and 2 (pg. 46-47), 1.

4 Computations and other Exercises

You are responsible for knowing any homework exercise that has been assigned as well as similar computational exercises. However, you may choose to focus on the following types.

  1. Determine if a set V is a vector space under specific operations.
  2. Determine if a subset W of a vector space V is a subspace of V.
  3. Determine if a subset S of a vector space V is a basis of V , and determine if a vector v ∈ span(S).
  4. Be able to do the following exercises from the textbook: Sec. 1.2: (9, 11), Sec. 1.4: (13, 15), Sec. 1.5: (12), Sec. 1.6: (12).

5 Proofs

You should be familiar with the various proof techniques we learned in class so that you will be able to do proofs and justifications similar to the homework problems. For example, how do you show that one set is contained in another set, how do you show that two sets are equal, or how do you show something is unique? We went over several different methods, so review them carefully. You should also be familiar with the proofs of each of the following theorems.

  1. Theorem 1.4: The intersection of any two subspaces of a vector space V is also a subspace of V.
  2. Theorem 1.5: The span of any subset S of a vector space V is a subspace of V. Moreover, any subspace of V that contains S also contains the span of S.

6 Good Luck!

Good luck on the exam! A lot of the work we have done has been computational, so make sure you can do all of the computational exercises from the homework. Make sure you practice enough of them so that the calculations can be done quickly (and correctly). The theoretical portion of the exam will probably be the more difficult aspect simply because the ideas are somewhat new to you. A lot of it is on this review sheet as well as the practice exam. Remember that the point of this course is really to help you become more comfortable with proofs because you should already know how to do the computations. With enough practice, you should be able to do fine.

You have all done well so far this semester. Keep up the good work, and don’t get stressed! Study thoroughly and you should be okay. Good luck!