Linear Algebra Study Guide for Exam 3

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10 Οκτ 2013 (πριν από 4 χρόνια και 29 μέρες)

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Linear Algebra


Study Guide for Exam
3




Exam 3 is on Chapter 4. We covered 4.1 thru 4.7.



I will NOT give you the theorems on this exam. You need to know them.
Most

were introduced in chapter 2 and should be very familiar to you.



You may use a graphing c
alculator. The only types I will not allow are those that
do symbolic manipulations (e.g., TI
-
89).



You will get a take
-
home part of the exam.



Section
4.1

You need to know the definitions for or how to do
/use

the following:



Prove or disprove a given set
is a vector space. This will be on the take home.



Check that a given set is a subspace of a vector space.



The span of a set of vectors forms a subspace.


Section
4
.2

You need to know the definitions for or how to do
/use

the following:



Definition of
the nu
ll space.



The null space forms a subspace of R
n
.



Given a matrix A, find the null space explicitly.



Definition of the column space.



The column space of A is a subspace of R
m
.



Know the fact in the blue box on page 230.



The definition for a linear transformat
ion on page 232.



Know the definition for the kernel and range of a linear transformation. This is
just below the green box on page 232 or get it from your lecture notes.


Section
4
.3

You need to know the definitions for or how to do
/use

the following:



The

definition of a linearly independent set which is on page 237 (equation 1).



Know theorem 4.



Definition of a basis for a subspace of a vector space (also is the definition for the
basis of the vector space itself). See page 238.



The Spanning Set Theorem o
n page 239.



How to find a basis for Nul(A) and Col(A).



Read “Two Views of a Basis” on page 242.


Section
4
.4

You need to know the definitions for or how to do/use the following:



Know the content of the Unique Representation Theorem (page 246). The
coordin
ates of a vector with respect to a given basis are unique.



The definition on page 246; just be familiar with the notation.



Know how to find the change of coordinates matrix. See equation 4 on page 249.



Theorem 8.





Section
4
.5

You need to know the defin
itions for or how to do
/use

the following:



Be clear on the content of theorems 9 and 10.



Know the definition of dimension on page 257.



Know theorems 11 and 12.


Section
4
.6

You need to know the definitions for or how to do/use the following:



Know the defin
ition for the row space of A. See page 263 or your notes.



Theorem 13.



Know how to find a basis for
Row (
A) given a matrix A.



Know the definition of rank; page 265.



The Rank Theorem.



The new entries in the Invertible Matrix Theorem on page 267.


Section
4
.
7

You need to know the definitions for or how to do/use the following:



Know how to find the coordinates of a given vector with respect to different
bases.



Theorem 15.


Section

4.8 and 4
.9

We skipped 4.8 and 4.9.



To study for the exam, know the above list

of topics. Do your homework. Review your
quizzes. Do the supplementary exercises on pp.
298
-
300

(odds are in the back of the
book).