Page 1
Some review questions for Linear Algebra final exam
Q1
Spring 2006 Math 22 Final Exam Review Sheet
You will not be allowed use of a calculator or any other device other than your pencil or
pen and some scratch paper. Notes are also not allowed. In kindness to your fellow test
takers, please turn off all cell phones and anything else that might beep or be a
distraction.
Be able to calculate:
• Echelon form of a coefficient or augmented matrix
• General solution to a linear system of equations
• Null space and column space of a matrix
• Kernel and range of a linear transformation involving polynomials
• Inverse of 2x2 matrix
• Eigenvalues and eigenvectors of 2x2 and 3x3 matrices (real or complex)
• How to diagonalize a square matrix, or in the case of complex eigenvalues, put
into the scaling plus rotation form
Know well and understand:
• Pivot columns and their meaning and uses
• Definitions of linearly independent and dependent, span, linear combination, one
toone, onto, linear transformation, vector space, subspace, null space, column
space, linearly independent, basis, dimension, rank
• Properties of invertible matrices (Invertible Matrix Theorem)
• Rank Theorem (p. 178)
• Matrix multiplication and rules with transpose and inverse (p. 115, 121)
• Matrix representation of a linear transformation
• Change of basis
• Orthogonality and the “perp” of a subspace
I would recommend reviewing each of these topics by skimming through the text and
your notes, then working through relevant homework problems, review exercises, and
exam problems. Finally, work the practice exam as though it were a true exam situation.
1. Let W=Span
!
1
1
0
0
"
#
$ $ $ $
%
&
' ' ' '
,
1
(1
2
1
"
#
$ $ $ $
%
&
' ' ' '
)
*
+ +
,
+ +

.
+ +
/
+ +
. Find a basis for W⊥.
2. Find bases for the kernel and range of the transformation T:P2P3 defined by
!
T(p(t)) = p(t) + 3tp(t) . Prove that T is linear. Is it onetoone? Is it onto? What is
the matrix representation of this transformation with respect to the bases {1,t, t2} and
{1,t, t2, t
3}?
Q2
c. False, since the set may not span H. A set of linearly independent vectors that
spans H is a basis. Also, a maximally large set of linearly independent vectors
(that is, adding any other vector in H to the set makes it dependent) is a basis for
H.
d. True, since if a vector in {v1, v2, v3} can be written as a combination of the
other two, then {v1, v2, v3, v4} must be a linearly dependent set.
Practice Final Exam
1. (16pt) Consider the matrix A=
!
1 2 3 4
1 0 1 0
2 4 5 6
"
#
$ $
%
&
' ' .
(a) Find a basis for the column space of A.
(b) Find a basis for the null space of A.
(c) Is b=[2 2 4]T in the column space of A? If so, write down the general solution
of Ax=b.
2. (16pt) Suppose an n×n matrix A satisfies A2=A. What are all of the possible eigenvalues of A?
To what fundamental space (e.g., column or null space of A) does the eigenspace for each
eigenvalue of A correspond?
3. (16pt) Prove that if W1 and W2 are subspaces of a vector space V, then (W1∩W2)⊥= W1
⊥+
W2⊥, where W1
⊥+ W2
⊥={v1+v2: v1∈ W1
⊥ and v2∈ W2
⊥}.
4. (16pt) Prove that the transformation
!
T :P2"R
3 defined by
!
T(p) =
!
p(1)
" p (1)
" " p (1)
#
$
% %
&
'
( ( is linear, one
toone, and onto.
5. (20pt) State whether each of the following is true or false. If it is true, briefly explain why it is
true. If it is false, then give a true statement and briefly explain why the original was incorrect
and why the new statement is correct. (You don’t need to prove anything, simply state known
results.)
(a) If A is an m×n matrix whose columns do not span Rm, then the equation Ax=b is
inconsistent for some b in Rm.
(b) The homogeneous equation Ax=0 has the trivial solution if and only if this equation has
at least one free variable.
(c) A linear transformation
!
T :Rn"R
m is completely determined by its effect on the
columns of the n×n identity matrix.
(d) If the columns of an m×n matrix A are linearly independent, then the columns of A
span Rn.
6. (16pt) What must be true about s and t if the matrix
!
A =
2 2 s
2 3 t
4 5 7
"
#
$ $
%
&
' ' is not invertible?
Q3
c. False, since the set may not span H. A set of linearly independent vectors that
spans H is a basis. Also, a maximally large set of linearly independent vectors
(that is, adding any other vector in H to the set makes it dependent) is a basis for
H.
d. True, since if a vector in {v1, v2, v3} can be written as a combination of the
other two, then {v1, v2, v3, v4} must be a linearly dependent set.
Practice Final Exam
1. (16pt) Consider the matrix A=
!
1 2 3 4
1 0 1 0
2 4 5 6
"
#
$ $
%
&
' ' .
(a) Find a basis for the column space of A.
(b) Find a basis for the null space of A.
(c) Is b=[2 2 4]T in the column space of A? If so, write down the general solution
of Ax=b.
2. (16pt) Suppose an n×n matrix A satisfies A2=A. What are all of the possible eigenvalues of A?
To what fundamental space (e.g., column or null space of A) does the eigenspace for each
eigenvalue of A correspond?
3. (16pt) Prove that if W1 and W2 are subspaces of a vector space V, then (W1∩W2)⊥= W1
⊥+
W2⊥, where W1
⊥+ W2
⊥={v1+v2: v1∈ W1
⊥ and v2∈ W2
⊥}.
4. (16pt) Prove that the transformation
!
T :P2"R
3 defined by
!
T(p) =
!
p(1)
" p (1)
" " p (1)
#
$
% %
&
'
( ( is linear, one
toone, and onto.
5. (20pt) State whether each of the following is true or false. If it is true, briefly explain why it is
true. If it is false, then give a true statement and briefly explain why the original was incorrect
and why the new statement is correct. (You don’t need to prove anything, simply state known
results.)
(a) If A is an m×n matrix whose columns do not span Rm, then the equation Ax=b is
inconsistent for some b in Rm.
(b) The homogeneous equation Ax=0 has the trivial solution if and only if this equation has
at least one free variable.
(c) A linear transformation
!
T :Rn"R
m is completely determined by its effect on the
columns of the n×n identity matrix.
(d) If the columns of an m×n matrix A are linearly independent, then the columns of A
span Rn.
6. (16pt) What must be true about s and t if the matrix
!
A =
2 2 s
2 3 t
4 5 7
"
#
$ $
%
&
' ' is not invertible?
Q4
3. Prove that the null spaces of A and ATA are identical for any mxn matrix A.
4. Rewrite the system
!
x " y + 2z " w = "1
2x + y " 2z " 2w = "2
"x + 2y " 4z + w =1
3x " 3w = "3
as a matrix equation Ax=b. Solve this system
using Gaussian elimination. Find bases for the column and null spaces of the matrix A.
5. Suppose that {u1, u2,…, un} is an orthogonal basis for Rn where each uk is a unit vector
and the nxn matrix A can be written A=c1 u1 u1T+…+ cn un un
T. Prove that A is
symmetric and has eigenvalues c1,…, cn.
6. State whether each of the following is true or false. If it is true, briefly explain why it is
true. If it is false, then give a true statement and briefly explain why the original was
incorrect and why the new statement is correct.
a. Asking whether the linear system corresponding to an augmented matrix [a1 a2 a3 b]
has a solution is equivalent to asking whether b is in Span{a1, a2, a3}.
b. R2 is a subspace of R
3.
c. A linearly independent set in a subspace H is a basis for H.
d. If {v1, v2, v3, v4} is a linearly independent set, then so is {v1, v2, v3}.
Partial solutions:
1. W⊥=Span
!
"1
1
1
0
#
$
% % % %
&
'
( ( ( (
,
"1/2
1/2
0
1
#
$
% % % %
&
'
( ( ( (
)
*
+ +
,
+ +

.
+ +
/
+ +
2. Kernel is zero subspace; range is spanned by {1+3t, t+3t2,t
2+3t
3}. The
transformation is onetoone but not onto (you should prove each of these). The
matrix representation is
!
1 0 0
3 1 0
0 3 1
0 0 3
"
#
$ $ $ $
%
&
' ' ' '
.
3. If Ax=0 then clearly ATAx=0. If A
TAx=0 then (Ax)
. (Ax)=x
TA
TAx=0 and so
Ax=0.
4. General solution is given by x=1t,y=2s,z=s,w=t (that is, z and w are free
variables).
5. Guess what the eigenvectors must be!
6. a. True, since the column space of A is the set of all vectors b for which Ax=b is
consistent (has at least one solution).
b. False, R2 is NOT a subspace of R
3. A set like {[x,y,0]: x,y real} is a two
dimensional subspace of R3. (This can be viewed as an embedding of R
2 into
R3.)
Q5
Stanford University UM51A
2. (12 points total, 4 points each) Consider the following linear system:
x1 + hx2 = 24x1 + 8x2 = k
(a) Find all values for h and k so that the system has no solution.
(b) Find all values for h and k so that the system has a unique solution.
(c) Find all values for h and k so that the system has infinitely many solutions.
[email protected] 3 EPGY OHS
Stanford University UM51A
2. (12 points total, 4 points each) Consider the following linear system:
x1 + hx2 = 24x1 + 8x2 = k
(a) Find all values for h and k so that the system has no solution.
(b) Find all values for h and k so that the system has a unique solution.
(c) Find all values for h and k so that the system has infinitely many solutions.
[email protected] 3 EPGY OHS
Stanford University UM51A
2. (12 points total, 4 points each) Consider the following linear system:
x1 + hx2 = 24x1 + 8x2 = k
(a) Find all values for h and k so that the system has no solution.
(b) Find all values for h and k so that the system has a unique solution.
(c) Find all values for h and k so that the system has infinitely many solutions.
[email protected] 3 EPGY OHS
Stanford University UM51A
2. (12 points total, 4 points each) Consider the following linear system:
x1 + hx2 = 24x1 + 8x2 = k
(a) Find all values for h and k so that the system has no solution.
(b) Find all values for h and k so that the system has a unique solution.
(c) Find all values for h and k so that the system has infinitely many solutions.
[email protected] 3 EPGY OHS
Page 2
Q6
Stanford University UM51A
4. (10 points total, 5 points each)
(a) Find the inverse of the matrix A =
2
41 �1 22 �3 31 �1 1
3
5.
(b) Find all values of k for which the matrix A =
2
41 2 43 1 6k 3 2
3
5 is invertible.
[email protected] 6 EPGY OHS
Stanford University UM51A
4. (10 points total, 5 points each)
(a) Find the inverse of the matrix A =
2
41 �1 22 �3 31 �1 1
3
5.
(b) Find all values of k for which the matrix A =
2
41 2 43 1 6k 3 2
3
5 is invertible.
[email protected] 6 EPGY OHS
Q7Cal Poly Fall 2009
9. (8 points) Let y =
76
�and u =
42
�. Write y as a sum of two orthogonal
vectors, one in Span(u) and one orthogonal to u.
Math 206 13 Final Exam
Q8
Cal Poly Fall 2009
10. Suppose that {u1,u2,u3} is an orthogonal set of vectors in R4, and u1 = 2,u2 = 3, and u3 = 4. Let
y = 2u1 � 5u2 + u3.
(a) (4 points) Find y.
(b) (4 points) Find y · u1.
Math 206 14 Final Exam
Cal Poly Fall 2009
10. Suppose that {u1,u2,u3} is an orthogonal set of vectors in R4, and u1 = 2,u2 = 3, and u3 = 4. Let
y = 2u1 � 5u2 + u3.
(a) (4 points) Find y.
(b) (4 points) Find y · u1.
Math 206 14 Final Exam
Q9
Cal Poly [email protected]
3. (4 points) Suppose that A is a 5⇥ 8 matrix such that
⇢
2
66664
70412
3
77775,
2
66664
�312�6
8
3
77775,
2
66664
1�2
031
3
77775
�
is a basis for the column space of A. Find p and q so that the following statementis true: The nullspace of A is a pdimensional subspace of Rq.
Math 206 4 Exam 2
Cal Poly [email protected]
3. (4 points) Suppose that A is a 5⇥ 8 matrix such that
⇢
2
66664
70412
3
77775,
2
66664
�312�6
8
3
77775,
2
66664
1�2
031
3
77775
�
is a basis for the column space of A. Find p and q so that the following statementis true: The nullspace of A is a pdimensional subspace of Rq.
Math 206 4 Exam 2
Q10
Cal Poly [email protected]
5. (6 points) Suppose that � is an eigenvalue of an invertible matrix A with corresponding eigenvector v. Determine whether v is an eigenvector of the matrixA + cIn, where c is a scalar. If so, what is the corresponding eigenvalue?
Math 206 6 Exam 2
Q11
Page 3
Cal Poly [email protected]
6. (8 points total) The trace of an n⇥n matrix A is the sum of the entries on the
main diagonal of the matrix, and is denoted tr(A). Let A =
a b
c d
�.
(a) (4 points) Show that the characteristic polynomial of A is
p(�) = �
2 � tr(A)� + det(A).
(b) (4 points) Suppose that A has two distinct, real eigenvalues �1 and �2.Show that
tr(A) = �1 + �2.
Math 206 7 Exam 2
Cal Poly [email protected]
6. (8 points total) The trace of an n⇥n matrix A is the sum of the entries on the
main diagonal of the matrix, and is denoted tr(A). Let A =
a b
c d
�.
(a) (4 points) Show that the characteristic polynomial of A is
p(�) = �
2 � tr(A)� + det(A).
(b) (4 points) Suppose that A has two distinct, real eigenvalues �1 and �2.Show that
tr(A) = �1 + �2.
Math 206 7 Exam 2
Q12
Review problems:
1. Suppose u, v, and w are vectors in R4. Can the span of u, v, and w be all of R
4? Explain why
or why not from the perspective of our linear systems theory.
2. Let A be a 5x3 matrix, let y be a vector in R3. Suppose Ay=z (what size is z?). Why can you
conclude that the system Ax=4z is consistent?
3. Consider the following system where c is an unknown number:
4 26
41284
03 22
=++
!=++
=++
czyx
zyx
zyx
a. Determine all values of c for which the system is consistent.
b. Determine all values of c for which the system has a unique solution.
c. Determine all values of c for which the system has infinitely many solutions,
and state the general solution.
4. Consider the following system of linear equations:
!
2x1
+ 4x2" 2x
3= 2
"5x1
+ x2
+ x3
=1
3x1" 5x
2+ x
3= "3
, which has
the reduced echelon form
!
1 0 "3/11 "1/11
0 1 "4 /11 6 /11
0 0 0 0
#
$
% % %
&
'
( ( (
.
a. Write each nonpivot column, including the 4th
column, of the original
augmented matrix as a linear combination of the pivot columns. Clearly state
the values of the coefficients for each linear combination.
b. Write down the general solution of this system.
c. For what vectors b can Ax=b be solved? Will solutions be unique?
Partial solutions:
1. Choose any vector b in R4. The vector equation xu+yv+zw=b corresponds to a
linear systems with four equations but only three unknowns. There can be at most
three pivots, so there will be some b for which we cannot solve the system, that is,
that b is not a linear combination of u, v, and w, and so they do not span all of R4.
2. Because a solution is x=4y.
3. Consistent for all values of c, unique solution iff
!
c " 3, infinitely many solutions
iff c=3.
4. 3rd
column=−
!
3
111
st column−
!
4
112
nd column (of original A matrix). Ax=b can be
solved for vectors b that are a linear combination of the first two columns of A, in
which case there will be a free variable and hence more than one solution.
Q13
Practice Exam:
1. (10pt) A scientist solves a nonhomogeneous system of ten linear equations in twelve
unknowns and finds that three of the unknowns are free variables. Can the scientist be
certain that, if the right sides of the equations are changed, the new nonhomogeneous
system will have a solution? Be specific in your answer and explain carefully.
2. (15pt) Let A be an invertible 44! matrix. Answer the following questions about A:
(a) Will the columns of A be linearly dependent, independent, or could be either?
(b) Can you determine the span of the columns of A? If so, state what the span is.
(c) If a solution to Ax=b exists, will it be unique? Explain.
3. (15pt) Let C be a
!
3" 5 matrix. Answer the following questions about C:
(a) Will the columns of C be linearly dependent, independent, or could be either?
(b) State a condition under which the span of the columns of C will be all of R3.
(c) If a solution to Cx=b exists, will it be unique? Explain.
4. (10pt) Suppose that A=UDVT where U is a nonsquare matrix satisfying U
TU=I, V is
invertible, and D is a diagonal matrix with nonzero entries along the diagonal (this is
called the singular value decomposition of A). Prove that TTAAA1)( ! =
!
(VT)"1D
"1UT by
substituting in for A on the left hand side and simplifying.
Partial solutions:
1. If we write the system as Ax=b, then A has 9 pivots. This implies that the columns of
A do not span all of R10
and so we expect that for some vectors c the system Ax=c will
not have a solution.
2. (a) Independent (b) R4 (c) Yes, unique.
3. (a) Dependent (b) 3 pivots or transformation is onto (c) No, at least 2 free variables
4. Be very careful, remembering that (AB)T=B
TA
T and (AB)
1=B
1A
1
Q14Review problems:
1. Find a matrix A such that the transformation
!
xa Ax maps
!
1
3
"
# $ %
& ' to
!
1
1
"
# $ %
& ' and
!
2
7
"
# $ %
& ' to
!
3
1
"
# $ %
& '
(hint: write this down as a linear system and solve for A). Is this transformation one
toone? Onto? What is the span of the columns of A? Are the columns linearly
independent? Is A invertible? Is so, find A1
.
2. Find a matrix A such that the transformation
!
xa Ax maps
!
1
3
"
# $ %
& ' to
!
1
1
"
# $ %
& ' and
!
2
7
"
# $ %
& ' to
!
3
3
"
# $ %
& '
(hint: write this down as a linear system and solve for A). Is this transformation one
toone? Onto? What is the span of the columns of A? Are the columns linearly
independent? Is A invertible? Is so, find A1
.
3. Suppose that A is a 5x3 matrix and that there exists a 3x5 matrix C such that CA=I3.
If for a particular b the equation Ax=b has at least one solution, prove that this
solution must be unique. Is A onetoone or onto? What can we conclude about the
columns of A?
Partial solutions:
1.
!
A ="2 1
4 "1
#
$ %
&
' ( ; yes;yes; R
2; yes;yes.
2.
!
A ="2 1
"2 1
#
$ %
&
' ( no;no;all multiples of
!
1
1
"
# $ %
& ' ;no;no.
3. Multiplying Ax=b by C leads to conclusion that x=Cb, implying that this is the
only possible solution. The transformation is onetoone but not onto. The
columns of A must be linearly independent, but since there are only three of them
they cannot span all of R5.