Dr. Byrne
Fall 2011
Math 237
Section 102
Linear Algebra Review for Exam 3
Exam Topics and Skills within Chapters 5, 6, 8 and 9
A) Chapter 5: Proving that a Transformation is or isn’t linear.
i) Definition of a linear transformation.
ii) Proof that a function is linear: see Quiz 8, Retake Quiz 8
iii) Prove that a function isn’t linear (Homework problem #50)
B) Chapter 6: Matrix representation of a linear transformation.
i) Given a linear transformation, find its matrix representation.
(Homework problems #1a, 37a, 40, 42a, 62)
ii) Common linear transformations: find the matrix that induces a rotation,
projection, dilation, contraction or composition of these.
(Worksheet, Quiz 9, Ch 6 #7, 39)
C) Chapter 5: Kernel and Image of Linear Mappings.
i) Definition of the image and kernel of a linear transformation.
ii) Given a linear transformation, find a basis for the image and the kernel.
Homework problems: Ch 5: 16 (worked solution), 61, 62, 63,
D) Chapter 6: Change of Basis
i) Find the matrix that will convert a vector with coordinates with respect to one
basis to coordinates with respect to another basis. Homework
ii) Find the matrix representation for the differential operator for a vector space
defined by basis of a finite set of functions.
Homework problems: Ch 6 #8, 43
E) Chapter 8: Determinants
i) Find the determinant of a matrix using cofactor expansion, be able to produce to
matrix of minors and the matrix of signed minors. (See Quiz 10).
ii) Find the determinant of a matrix using row reduction. (Will need to show
intermediate steps.)
iii) Singular matrices: Know that the determinant is 0 if and only if the rank of A is
less than n. (That is, rows are linearly independent.) Be familiar with other
equivalencies.
F) Chapter 9: Eigenvalues and Eigenvectors
i) Derive the characteristic polynomial from first principles. (Quiz 11)
ii) Given a matrix, find the eigenvalues and eigenvectors.
(Homework: 9a, 45, 46a, 48)
iii) Given a matrix, determine if it is diagonalizable and diagonalize it.
(Homework: 9, 45, 46, 48) (this will be on the final, though not on Exam III)
iv) Use diagonalization to find An.
Linear Algebra Practice Exam 3
(Problems I have had on previous exams. Sections C and D are new with this book so
these problems are not represented below.)
Chapter 5 and 6 Linear Transformations
1 1 
0 
(1) Let A = 1 2 . What is the image of x=   under the transformation T: x  Ax?


1 
1 3
1 1
1
1 2  0  2 The image is

 1  
1 3   3
1 
2 .
 
 3 
(2) Show that the transformation T:    defined by T ( x)  x is not linear.
Property 1 of linearity: T(u)+T(v)=T(u+v)
The transformation T fails property 1 if u=4 and v=9:
T (u )  4  2
T (v )  9  3
so T (u )  T (v)  2  3  5
T (u  v)  13
T (u  v)  T (u )  T (v) since 13  5
(3) For each part below, consider transformations T: 2  2 (i.e., functions that map
a vector from 2 to 2.)
a) Find a matrix A such that left multiplication by A dilates a vector by a factor of
3 in all directions.
b) Find a matrix B such that left multiplication by B projects a vector onto the yaxis.
c) Find a matrix C such that left multiplication by C dilates a vector by a factor of
3 in all directions and then projects it onto the y-axis.
 x1 
 x   x1  x3 
4
3
(4) Define T :    by T (  2  )   x 2  x 4  .

 x3  
   x1  x3 
 x4 
Find a matrix A such that T is left
multiplication by A.
Find the image of the basis vectors:
1
0 
0
0 
0 1
1 0
0 1
0  0 






T (e1 )  T (   )  0 , T (e 2 )  T (   )  1 , T (e3 )  T (   )  0 , T (e 4 )  T (   )  1
0 
0 
1
0 
  1
  0
  1
  0
0 
0 
0
1
1 0 1 0
So AT  T (e1 ) T (e 2 ) T (e3 ) T (e 4 )  0 1 0 1 .
1 0 1 0
Chapter 8: Determinants
5. Find the inverse of A using A-1 =
1
adj ( A) .
det( A)
0 1 1 
1 0 1

A= 
1 1 0
6. For the matrix A,
a) Show |A|=35 using expansion by cofactors.
b) Show |A|=35 using row reduction.
1
0

0

A  0
1

0
0

2 3 4 5
1 0 0 0
0 5 0 0
0 0 1 0
0 3 0 6
0 0 0 0
0 0 0 0
7
 1 0
0 0

0 0
2 0

7 0
0 1 
6
7. In a linear system Ax  B , A is an n  n square matrix with rank n. How many
solutions may the system have?

If the system is consistent, it has exactly one solution. (There are no free variables.)
However, the system may be inconsistent (if B is not in the column space of A) in
which case there will be no solution.
 In contrast, the system Ax  0 has exactly one solution because the trivial solution
is always a solution for a homogeneous system.
Chapter 9: Eigenvalues and Eigenvectors
8. If A is an n x n matrix,  is an eigenvalue for A if and only if p()=det(A-I)=0.
Explain the origin of the characteristic polynomial p() by deriving in detail that
p()=det(A-I).
Apply definition:  is an eigenvalue
implies
Ax  x
for some nonzero vector x.
Ax  x
Ax  x  0
Ax  Ix  0
 A  I x  0
The linear system  A  I x  0
has a non-trivial solution only if det  A  I   0 .
9. Answer part (a) and (b) below for an n x n matrix A.
a) Write the definition of an eigenvector of A.
b) Find the eigenvalues of A =
6 3  8
0  2 0 .


0 0  3
10. Answer part (a) and (b) below.
6 3  8
a) The eigenvalues of A = 0  2 0  are 6, -2 and -3. Find the eigenvectors


0 0  3
and eigenspaces of A.
b) Without diagonalizing A, determine if A is diagonalizable. Explain your
reasoning.
A is diagonalizable
because it has 3 linearly independent eigenvectors
(3 are needed since A is 3x3).
2
2
4
1 0


11. The eigenvectors of A=  5  3  2 are  1 ,  1 ,


   
 5
 1   1 
5
4 
 1
 1 .
 
 0 
a) Write down a matrix P that diagonalizes A.
b) Given the P you wrote down in part (a) above, write down D, the
diagonalization of A where D=P-1AP.
 3  2
12. Find the complex eigenvalues of A  
.
4  1 
 3  2
13. Find 

4  1 
10
In case you’re curious, this is the solution:
14. Given the following matrix A:
 7 15
A

 0 8
a.
Find a diagonalizing matrix P of A such that
diagonal matrix.
P 1 AP  D
where D is a
5 points: characteristic polynomial det A  I   0
 7 15  0   7   15
0
 0 8   0   0
8

 

since the matrix is upper triangular:
(7   )(8   )  0
5 points: finding the roots to the characteristic polynomial
 = -7, 8
5 points: finding the eigenvectors
0 15  x  0
0 1  x  0
 A  I x  
   



      by row reduction.
0 15  y  0
0 0  y  0
 x
1
1 
 = -7:
 y   s 0 Only 1 eigenvector: 0 
 
 
 
 15 15  x  0
 A  I x  

  15 x  15 y  0  x  y
0   y  0
 0
 = 8:
 x
1
1
 y   s 1 Only 1 eigenvector: 1
 


5 points: Constructing P from the eigenvectors
1 1
P

0 1
b. Show that if P
1
AP  D then A  PDP 1 .
P 1 AP  D
PP 1 AP  PD
AP  PD
APP 1  PDP 1
APP 1  PDP 1
A  PDP 1