Some review questions for Linear Algebra ﬁnal exam Q1 Let W=Span 1 1 0 0 " # \$ \$ \$ \$ % & ' ' ' ' , 1 (1 2 1 " # \$ \$ \$ \$ % & ' ' ' ' ) * + + , + + - . + + / + + . Find a basis for W . Q2 (16pt) Consider the matrix A= 1 2 34 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. Q3 (16pt) Prove that the transformation T : P 2 " R 3 defined by T ( p) = p(1) " p (1) " " p (1) # \$ % % & ' ( ( is linear, one- to-one, and onto. Q4 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 [a 1 a 2 a 3 b] has a solution is equivalent to asking whether b is in Span{a 1 , a 2 , a 3 }. b. R 2 is a subspace of R 3 . c. A linearly independent set in a subspace H is a basis for H. d. If {v 1 , v 2 , v 3 , v 4 } is a linearly independent set, then so is {v 1 , v 2 , v 3 }. Q5 Consider the following linear system: x 1 + hx 2 = 2 4x 1 +8x 2 = 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.
3

# Spring 2006 Math 22 Final Exam Review Sheetlegacy.earlham.edu/~pardhan/courses/linear/tq/[email protected] 3 EPGY OHS Stanford University UM51A 2. (12 points total, 4 points

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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-

to-one, 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 one-to-one? 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-

to-one, 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-

to-one, 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 one-to-one 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=-1-t,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

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 p-dimensional 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 p-dimensional 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 cor-responding 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

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 non-pivot 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

!

(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 non-square 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-

to-one? Onto? What is the span of the columns of A? Are the columns linearly

independent? Is A invertible? Is so, find A-1

.

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-

to-one? Onto? What is the span of the columns of A? Are the columns linearly

independent? Is A invertible? Is so, find A-1

.

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 one-to-one 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 one-to-one 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.