Math 410 Homework Problems

In the following pages you will find all of the homework problems for the semester. Homework

should be written out neatly and stapled and turned in at the beginning of class on the due date

specified by your instructor. No late homework will be accepted! Each homework will be worth

10 points. There are many challenging problems to be found here so don’t be upset if you are not

able to do all of them on each and every assignment, though you should try to complete them all.

However, please keep in mind that turning in rubbish on a problem or trying to fake that you

attempted to do a problem does not reflect well on you as a student and can be frustrating. Also,

there is not enough class time to answer homework questions in class, but I expect that you will

come and talk to me if you need some help or hints on the problems and we can even have a

regular meeting time for homework help. Some of the homework has an asterisk by the problem,

which indicates that the problem is extra challenging or requires some diligence on your part,

and a double asterisk problem will likely require significant exploration. However, you should

not automatically skip the asterisk problems and in fact, many times when you look at a

challenging problem in the correct way, the problem is actually easy. Also, please keep in mind

that the asterisk problems are by no means the only challenging problems, so don’t feel badly if

you find many of the problems without an asterisk challenging too. Remember that the challenge

and struggle is an indication that you are stretching yourself to grow in your intellectual abilities,

which is the primary reason you are going to college. If you challenge yourself and you will be

amazed at how much you are capable of doing! Homework will be graded on both completeness

and on certain problems which will be chosen to grade for correctness. In general, just doing the

first few problems in each section, incomplete work or turning in rubbish on the problems will

result in a low homework score. When it comes to linear algebra, to be successful you must put

in a lot of time and diligence on homework and successful students often come by office to

discuss the homework problems several times each week.

“The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it

forces on the would be solver” I.N. Herstein

“Very often in mathematics the crucial problem is to recognize and discover what are the relevant

concepts; once this is accomplished the job may be more than half done.” I.N. Herstein

Chapter 1 Homework for Exam 1 Linear Systems and Matrices

Section 1 Basics of Linear Systems and Equations

Problem 1 Determine which equations are linear and which are not linear.

I II

√ = 4 III √

IV

V

Problem 2 Determine whether the equations form a linear system

I {

II {

√

III {

Problem 3 Determine which vectors are solutions of the system

{

I II III (

) IV (-1, -1, -1 )

Problem 4 In each part, find a system of equations corresponding to the augmented matrix

I [

] II [

] III [

]

Problem 5 Find the augmented matrix for each system

I {

II {

III {

IV {

Section 2 Gaussian Elimination

Problem 1 Which of the following matrices are in reduced row-echelon form?

I [

] II [

] III [

] IV [

]

Problem 2 Which of the following matrices are in row-echelon form?

I [

] II [

] III [

] IV [

]

Problem 3 Solve each linear system by Gauss-Jordan elimination.

I {

II {

III {

IV {

V { VI {

VII {

Problem 4 Solve each homogeneous system

I {

II {

III {

IV {

V {

VI {

Problem 5 Suppose that the row-echelon form of a system with four unknowns is given

by the augmented matrix below. Then write the parametric solution for the system.

[

]

Problem 6 Determine the values of a for which the system has a unique solution, infinitely many

solutions or no solutions.

I {

II {

Problem 7 Find the equation of the circle that passes through the points

Problem 8 Find the constants such that the graph of the equation

passes through the points

*Problem 9 Suppose a linear system of 4 equations in 4 unknowns has two distinct solutions. Prove that

the system must have an infinite number of solutions.

True or false

State whether each of the following statements are true or false. If the statement is true explain why it is

true. If the statement is false, give an example to show that the statement is false for that example.

(a) A homogeneous system is always consistent.

(b) If a system of equations has more than one solution, then it must have an infinite number of

solutions.

(c) If are two distinct solutions of a linear system with three equations

in three unknowns then is also a solution of that same system for

any real number

(d) A homogeneous system always has an infinite number of solutions.

Section 3 Arithmetic of Matrices and Matrix operations

Problem 1 Let [

] [

]. Compute each expression

(a) (b) (c) (d)

Problem 2 Let [

] [

] . Then find

Problem 3 Let [

] and [

]. Then find each of the following:

(a) (b)

Problem 4 In each part, find matrices and write the system as a matrix equation

(a) {

(b) {

Problem 5 Find the values a, b, c and d:

[

] [

]

Problem 6 Find the value of k that makes the equation true:

[

] [

] [

]

Problem 7 Let be a that satisfies the matrix equation [

] [

]. Find

.

Problem 8 For each 3 x 3 matrix [ ] find

(a) {

(b) {

(c) {

Problem 9 Let [ ] be a 3 x 3 matrix such that {

and define

Then find the matrix .

Problem 10 Prove that if .

Problem 11 Prove that if

Section 4 The Inverse of a Matrix and Properties

Problem 1 Le [

] [

] [

] and .

Then verify the following properties for these given matrices:

I II III IV .

Problem 2 Let [

] [

] [

] [

]. Find the

inverse of each of these matrices.

Problem 3 Let [

] [

] Then verify the following properties for these given

matrices.

I II

Problem 4 In each of the following, use the given information to find the matrix

I [

] II [

] III ( [

])

Problem 5 [

] and . Find

*Problem 6 Let [

] and

Show that [

] (Hint: this problem may take a while to do the calculation)

Problem 7 Suppose that . Then using the

given information and properties of matrices show that

[

].

Problem 8 Suppose that .

Then by correctly using the given information and properties of matrices show

Problem 9 Show that if is a square matrix that satisfies then is invertible and find .

Problem 10 Show that if is a square matrix that satisfies then is invertible and find

.

Problem 11 Show that if is a square invertible matrix and then is its own inverse.

Problem 12 Suppose is a square matrix that satisfies . Show that and

is invertible and its own inverse.

*Problem 13 Suppose that is a square matrix that satisfies for some positive integer n. Show

that is invertible and .

Problem 14 Prove that if is an invertible matrix then is invertible and .

Problem 15 Prove that if are invertible matrices with the same size then is invertible and

*Problem 16 If find all powers of that are equal to the inverse of

Problem 17 Show that matrix containing a row of zeros then is singular.

Section 5 Elementary Matrices and Inversion

Problem 1 Determine whether each of the given matrices are elementary matrices.

I [

] II [

] III [

] IV [

] V [

]

Problem 2 What elementary row operation is needed to make each elementary matrix the identity matrix?

I [

] II [

] III [

] IV [

] V [

]

Problem 3 Suppose that

[

] [

] [

] [

].

For each equation, find an elementary matrix that satisfies the equation.

I II III

Problem 4 For each matrix, use the inversion algorithm to find the inverse of the matrix, if the inverse

exists.

I [

] II [

] III [

] * IV [

]

Problem 5 For find the inverse of the matrix [

].

Problem 6 Write each matrix as a product of elementary matrices.

I [

] II [

] *III [

]

Problem 7 Let be an invertible matrix, an matrix and column vector

(or matrix) so that the matrix equation represents a homogeneous system of equations in

unknowns. Show that if has just the trivial solution, then is invertible.

Problem 8 Prove that if is an invertible matrix and row equivalent to then is invertible.

Problem 9 Prove that if matrices then are row

equivalent to each other.

Problem 10 Let an matrix and column vector (or matrix) so that the matrix

equation represents a homogeneous system of equations in unknowns. Show that if

has only the trivial solution then has only the trivial solution.

Problem 11 Suppose that column matrix (or

column vector). Prove that if is an invertible matrix then has only the trivial solution,

[

] .

Section 6 Using the Inverse of a Matrix to Solve Systems

Problem 1 Solve each system by inverting the coefficient matrix

I ) {

II {

III {

IV {

Problem 2 Solve the linear systems together using an augmented matrix

I {

(i) (ii)

II {

(i) (ii)

Problem 3 Find the value of so that the system I {

is consistent.

Problem 4 Show that for the system to be consistent {

,

we must have .

Problem 5 Suppose that [

] [

].

I Then solve the homogeneous system and use the result to solve [

]

II Solve

Section 7 Special Matrices

Problem 1 Determine by inspection whether each of the diagonal matrices is invertible. For those that

are invertible, find the inverse.

I [

] II [

] III [

] IV [

]

Problem 2 Find each product.

I [

] [

] II [

] [

] III [

] [

]

Problem 3 For each matrix find

I [

] II [

] III [

]

Problem 4 Determine which of the following matrices are symmetric

I [

] II [

] III [

] IV [

]

Problem 5 Find all values of so that the given matrix is symmetric

[

]

Problem 6 Determine which of the following matrices is invertible

II [

] III [

] IV [

]

Problem 7 Find all diagonal matrices such that .

Problem 8 Prove that if is an matrix then is a symmetric matrix.

Problem 9 Prove that if is an matrix then is a symmetric matrix.

*Problem 10 Prove that if then (Hint: if you look at this the correct way the problem

is easy otherwise you may get stuck)

Problem 11 Prove that if is a symmetric matrix then

I is a symmetric matrix and II is a symmetric matrix.

Problem 12 For each 3 x 3 matrix [ ] determine whether is symmetric

I II III IV

Problem 13 We say a matrix is skew-symmetric if

I Show that if is invertible and skew-symmetric then is skew-symmetric.

II Show that the matrix is skew-symmetric.

III If is an matrix, then can be expressed as the sum of a symmetric matrix and a skew-

symmetric matrix (hint: Use

)

Problem 14 Let [

]. Then show that is skew symmetric.

Problem 15 Suppose that [ ] , where Prove that

is skew symmetric.

Problem 16 Prove that if [ ] and is both symmetric and skew symmetric then is an

matrix with all zero entries.

Chapter 1 True/False Quiz

State whether each of the following statements are true or false. If the statement is true explain why it is

true. If the statement is false, give an example to show that the statement is false for that example.

1. A linear system whose equations are all homogeneous must be consistent.

T F

2. The linear system

Cannot have a unique solution, regardless of the value of

T F

3. Elementary row operations permit one equation in a linear system to be subtracted from

another.

T F

4. If a matrix is in reduced row echelon form then it is also in row echelon form.

T F

5. If a matrix is in row echelon form and in every column with a leading one all other entries in the

column other than the one are zero, then the matrix is in reduced row echelon form.

T F

6. If are two matrices with the same size, then .

T F

7. If are two matrices with the same size, then the product is always defined.

T F

8. If are 2 x 2 matrices and are both invertible matrices, then

T F

9. If an elementary row operation is applied to a matrix that is in row echelon form, then the

resulting matrix will still be in row echelon form.

T F

10. If has a column of zeros, then so does , if this product is defined.

T F

Chapter 2 Homework for Exam 2 Determinants

Section 1 Definition of Determinants and Cofactors

Problem 1 Let [

]. Find the minors and corresponding cofactors for .

*Problem 2 Let

[

]

. Find .

Problem 3 Find the determinant and the inverse of each matrix with nonzero determinant.

I [

] II [ √ √

√ √ ] III [

]

Problem 4 Evaluate the determinant of each matrix by cofactor expansion along a row or column

I [

] II [

] III [

]

IV [

]

Problem 5 Solve the equation for :

| [

] [

]|

Problem 6 Solve for |

| |

| |

| |

|

Problem 7 Solve each equation for

I |

| II |

|

Problem 8 Solve for |[

] [

]|

Problem 9 Find the value of each determinant by inspection.

I |

| II |

| III |

|

Problem 10 Show that if |

| with [

] [

], then

*Problem 11 Show that if [

], then

|

|

Section 2 Determinants and Row Reduction

Problem 1 Find the value of each determinant by inspection.

I |

| II |

| III |

|

Problem 2 Evaluate each determinate by using row operations and reducing the matrix to row-

echelon form.

I |

| II |

| III |

|

IV ||

|| V |

| *VI ||

||

Problem 3 Given that |

| , find the value of each determinant

I |

| II |

| III |

|

*Problem 4 Use row or column operations to find the determinant in factored form (you must show all

work not just state the answer):

|

|

Problem 5 Factor |

| |

| |

| completely

using properties of determinants.

Problem 6 Suppose that [

] and [

], where is a

constant. Prove that

Problem 7 Suppose that [

] and [

] i.e., we obtain

by interchanging the first two rows of Prove that

Problem 8 Use mathematical induction to prove that for any positive integer and any matrix

Hint: Let { }. Show that

by first showing that and then assume (your induction hypothesis) and use the

induction hypothesis to show must also be in . Conclude by the Principle of Mathematical

Induction (PMI) that is equal to the set of all positive integers , which proves the result.

Section 3 Properties of Determinants

Problem 1 Write a statement explaining how you use the determinant of a square matrix to

determine whether the matrix is invertible.

Problem 2 Using determinants determine which of the following matrices are invertible (see

Problem 1):

I [

] II [

] III [

]

Problem 3 Find the value of so that the matrix is not invertible

[

]

Problem 4 Show that [

] is invertible for all real values of and find

Problem 5 Find all values of such that the matrix [

] is nonsingular.

Problem 6 Let

Find the value of

I II III

Problem 7 Let

Find the value of

I II III

Problem 8 Using determinants, prove that if are invertible matrices then is

an invertible matrix.

Problem 9 Using determinants, prove that if are invertible matrices,

where then is an invertible matrix.

Problem 10 Show that if is any square matrix such that for some positive integer

then is not invertible.

Problem 11 Prove directly using the fact that an invertible matrix can be written as a product of

elementary matrices and properties of determinants, that if

Problem 12 Prove that if a square matrix is invertible then is invertible.

Problem 13 Prove that if a square matrix such that is invertible, then is invertible.

Problem 14 Let such that is invertible and . Show

that .

Problem 15 Suppose that is an invertible matrix. Prove that

⁄

Section 4 Using Determinants to Solve Systems-Cramer’s Rule

Problem 1 Use Cramer’s Rule to solve each system of equations

I {

II {

III {

* IV {

Problem 2 Use Gaussian-Jordan elimination to prove Cramer’s Rule in the case of two linear

equations in two unknowns i.e., show that if {

is a system of two equations in

two unknowns such that |

| then

|

|

|

|

|

|

|

|

Problem 3 Find the adjoint of each matrix

I [

] II [

]

Problem 4 Prove that if is a square matrix with integer entries and , then

contains all integer entries.

*Problem 5 Prove that if matrix then ( ) .

Problem 6 Let Prove that ( ) .

*Problem 7 Let . Show that iff

Problem 8 Use Problem 7 to show that if

*Problem 9 Prove that if matrix then ( ) .

Problem 10 Prove that then is invertible and

Problem 11 Let [

]. Find the value of

Problem 12 Let

Find the value of

I II III

Problem 13 Suppose that [

] and

[

] , where is a constant. Prove that

Problem 14 Let [

]. Find

Problem 15 Suppose that ( )

and Find the exact numerical value of .

State whether each of the following statements are true or false. If the statement is true explain why it is

true. If the statement is false, give an example to show that the statement is false for that example.

1. Suppose . Then for all such matrices .

T F

2. The determinant of an elementary matrix must be equal to either 1 or -1.

T F

3. Cramer’s rule can be used to find the solution to any consistent linear system of

T F

4. Cramer’s rule can never be used to solve a homogeneous linear system.

T F

5. For any invertible matrix that contains only integer entries, will contain only integer

entries.

(Hint: consider the formula

)

T F

6. For any is invertible.

T F

7. If [ ]

then

T F

8. For all 2 x 2 matrices such that

T F

9. For all that are not inverses of each other,

T F

10. is a square matrix with a column of zeros, then there exists a square matrix

.

T F

Chapter 3 Homework for Euclidean Vector Spaces

This chapter will provide a review of material that is often covered in Math 402 or Calculus III.

Those students who have not taken Math 402 should plan to spend more time on this material.

Section 1 Euclidean and Basic Vector Properties

Problem 1 Sketch the vector

I II .

Problem 2 Given the points sketch the vector with its initial point at the origin and

write the vector in component form.

(a) (b) (c)

Problem 3 Find a unit vector that has the same direction as

Problem 4 Let Then write each vector in

component form

(a) (b) (c)

Problem 5 Find a value of such that the given vector is parallel to

(a) ( √ ) (b) (

√ )

Problem 6 Perform the indicated vector arithmetic in when

.

(a) (b) (c)

Section 2 Dot products and Norms of Vectors

Problem 1 Which of the following is a unit vector?

(a) (b)

(c) (d) (

√ ⁄

√ ⁄ )

Problem 2 Find the norm of each given vector

(a) (b) (c)

Problem 3 Let Find

I II III IV

Problem 4 Find the Euclidean distance between the given vectors.

I II III

IV V

Problem 5 suppose that ‖ ‖ ‖ ‖ √ . Find

(a) (b)

Problem 6 Prove that if are vectors in , such that then

‖ ‖

‖ ‖

is a unit vector in

Problem 7 Find a unit vector having the same direction as the given vector .

(a) (b) (c) (d)

Problem 8 Prove that for any vectors .

*Problem 9 Prove that for any vectors |‖ ‖ ‖ ‖| ‖ ‖.

Problem 10 Prove that for any vectors ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ .

Problem 11 Let Describe the set of all points

‖ ‖ by identifying the type of conic and finding its equation.

Problem 12 Let Describe the set of all points

‖ ‖ ‖ ‖ by identifying the type of conic and finding its

equation.

Problem 13 Describe the set of all points such that ‖ ‖ by

identifying the geometric solid that is the solution set of the inequality.

Problem 14 Given vectors with ‖ ‖ ‖ ‖ find the largest

possible value for ‖ ‖.

Section 3 Angle and Orthogonality

Problem 1 Find the angle between the given vectors

I II

III

Problem 2 Determine whether the given pair of vectors are orthogonal.

I II

III IV

Problem 3 Determine whether the given set of vectors is an orthogonal set.

I { } II { } III { }

IV { } V { }

Problem 4 Find a unit vector that is orthogonal to .

Problem 5 Find the point normal form for the equation of the line passing through

having normal .

Problem 6 Find the point normal form for the equation of the plane passing through

having normal .

Problem 7 Determine if each given set of planes is parallel.

I

II

Problem 8 Given find and ‖ ‖

I II

III IV

Problem 9 Find the vector component of and the vector component of

orthogonal to and sketch the graph of both.

I II

Problem 10 Prove that for any nonzero vectors the vector is orthogonal to

.

Problem 11 Determine whether each given pair of planes is perpendicular

I

II

Problem 12 Find the distance between the point (-3,2) and the line determined by

.

Problem 13 Find the distance between the point each given point to the plane

(a) (b) (c)

Problem 14 Find the distance between the parallel planes determined by the equations

*Problem 15 Prove that if { } and

we have scalars such that , then each of the scalars is equal to

zero.

*Problem 16 Suppose that are nonzero vectors in and let ‖ ‖, ‖ ‖ .

Then if , show that the vector bisects the angle between

Problem 17 Suppose that and is orthogonal to each of the vectors

in . Then prove that is orthogonal to for any

scalars .

Problem 18 Suppose that Prove that

‖ ‖ ‖ ‖ ‖ ‖ .

Problem 19 Equations and

produce parallel planes in with ‖ ‖ and

‖ ‖ . Show that | | .

Section 4 Cross Products and Parametric Curves in

Problem 1 Find the vector and parametric equation of the line containing point that is

perpendicular to the vector and also write the equation of the line in slope-intercept

form .

Problem 2 Find the vector and parametric equation of the line containing that is

parallel to the vector with initial point and terminal point

Problem 3 In each part, use the given parametric equation of the line to find a point on the line

and a vector parallel to the line.

(a) (b)

Problem 4 Write the vector equation of the plane with equation .

Problem 5 If the vector equation of a plane is where

then write the equation of the plane in the form

.

Problem 6 Find the vector equation of the plane containing the point (1,-2,5) and the parallel

vectors

Problem 7 Use the cross product to find a vector that is orthogonal to both

I II III

Problem 8 Find the equation of the plane containing the point (2,-1,1) and the parallel vectors

in the form by using the cross product to

find the normal.

*Problem 9 Two planes in with vector equations

intersect in a line. Write the parametric equations for the line of

intersection of the planes (Hint: eliminate the parameters and solve a system).

Problem 10 Let and consider the equation

which has a graph that is a plane that passes through the origin in

Show that if then for any two scalars

is also a vector in the plane (Hint: use the parametric form of the

plane).

Chapter 4 Homework for Vector Spaces

In this chapter will look a generalization of which we call real vector spaces. This is where

the fun begins.

Section 1 Basics of Real Vector Spaces

Problem 1 Let { }. Show that is a real vector space using the standard

operations on .

Problem 2 Let { }. Show that is a real vector space

using the standard operations on .

Problem 3 Let { }.

Show that is a vector space under the standard operations of polynomial addition and

multiplication of a polynomial by a scalar.

Problem 4 Let { [ ] } with the usual operations of function addition and

multiplication of a function by a scalar. Show that is not a vector space by giving a vector

space axiom that fails to hold.

Problem 5 Let Let { }. Show that is a

real vector space under the standard operations in

Problem 6. Prove that the set of all points on a line in is a vector space if the line passes

through the origin.

Problem 7 Explain why a line in that does not pass through the origin is not a vector space

by exhibiting a vector space axiom that is not satisfied.

Problem 8 Let { }. Show that is a vector space.

Section 2 Subspace of a Vector Space

It is easier to show that a subset of a vector space is a vector space since we need only satisfy the

three hypotheses of the Subspace Theorem.

Hypotheses:

1. The set is not empty.

2. The set is closed under addition.

3. The set is closed under scalar multiplication.

Conclusion: The subset is a vector space so it is a subspace.

Problem 1 Show that if is a

subspace of under the vector space operations in

Problem 2 Show that { } is a subspace of .

Problem 3 Show that {[

] } is a subspace of .

Problem 4 Let be nonzero vectors in and { }.

Show that is a subspace of

Problem 5 Let be a nonzero vector in and let { }. Show that is a

subspace of .

Problem 6 Which of the following is in the span of { }

(a) 1 (b) (c) (d) 0

Problem 7 Let Show that if

then

Problem 8 Let { }. Is a subspace of ? If your answer is “yes”

then prove that is a subspace of . If your answer is “no” then give a vector space axiom that

does not hold and a specific example where the axiom does not hold.

Problem 9 Show that { } is .

Problem 10 Let Show that { [ ] } is a subspace

of [ ]

Problem 11 Let { }.

Show that is a subspace of

Problem 12 Show that { [ ] ∫

} is not a subspace of [ ] by

exhibiting a counter example.

Problem 13 Express as a linear combination of the vectors ,

and

Problem 14 Determine whether the vectors , span

Problem 15 Let [

]. Show that the solution set of the homogeneous system

is a line that goes through the origin in .

Problem 16 Show that the solution set of a consistent nonhomogeneous system of 3 equations

in 3 unknowns does not form a subspace of .

Problem 17 Let a vector space and let { } be a subset of and also

{ } a subset of Show that if and only if each vector

in can be written as a linear combination of the vectors in and each vector in can be

written as a linear combination of the vectors in

Section 3 Linear Independence

Suppose that and { } is a finite subset of . Then we say

that the set { } is linearly independent if whenever there are constants

such that then

In short, linearly independence means

Problem 1 Which of the following sets of vectors from is linearly independent?

I { } II { } III { }

IV { } V { }

VI { } VII { }

Problem 2 Which of the following sets of vectors from is linearly independent?

I { } II { } III { }

IV { } V { }

VI { } VII { }

Problem 3 Show that if { } is a linearly independent subset of a vector space

then { } is a linearly independent subset of

Problem 4 Show that if { } is a linearly dependent subset of a vector space then

{ } is a linearly dependent subset of

Problem 5 Use the dot product to show that if { } is an orthogonal set of

nonzero vectors in then is a linearly independent subset of .

Problem 6 Suppose { } is a linearly independent subset of a vector space

and not in Show that { } is linearly

independent.

Problem 7 Suppose { } is a subset of a vector space and is a vector with

Show that { } is linearly dependent.

Problem 8 Suppose that { } is a set of vectors from a vector space . Prove that if

the vector then { } is a linearly dependent subset of

Problem 9 Suppose that { } is a set of vectors from a vector space Prove that the set

{ } is a linearly dependent subset of

Problem 10 Use the Wronski’s Test to show that { } is a linearly

independent subset of .

Problem 11 Use the Wronski’s Test to show that { } is a linearly

independent subset of .

Problem 12 Use the Wronski’s Test to show that { } is a

linearly independent subset of .

Section 4 Basis and Dimension

Problem 1 Using pencil and paper, define what it means for a set of vectors to be a basis for a

vector space.

Problem 2 Using a pencil and paper define what it means for a vector space to be finite

dimensional.

Problem 3 Circle each of the following vectors spaces that are not finite dimensional.

I II III [ ] IV V { [ ] [ ]}

Problem 3 Determine whether the given set is a basis for the given vector space and be sure to

explain your answers.

a. {(

) (

) }

; space

b. { } space

Problem 4 Which of the following subsets of is a basis for ? Be sure to justify answers.

I { } II { } III { }

Problem 5 Which of the following is a basis for ? Be sure to justify answers.

I { } II { } III { }

Problem 6 Which of the following is a basis for ? Be sure to justify answers.

I { } II { }

Problem 7 Find a basis for the solution space of the homogeneous system:

{

Problem 8 Find a basis for the solution space of the homogeneous system:

{

Problem 9 Find a basis for subspace of determined by the line .

Problem 10 Find the dimension of each of the following vector spaces:

I The vector space of all diagonal matrices.

II The vector space of all skew symmetric matrices.

III The vector space of all lower triangular matrices.

IV The vector space of all symmetric matrices.

Problem 11 Consider the subspace of given by { }. Find the dimension

of this subspace.

Problem 12 Let { }. Show that is a subspace of and find the

dimension of this subspace.

Problem 13 Suppose that { } is a basis for a 4-dimensional vector space with

and constants such that Show that if

are also constants such that , then

, , , .

Problem 14 Let and . Find a vector that can be added to the

set { } to produce a basis for .

Problem 15 Let {[

] [

] [

] [

]} Show that is a basis for

. Then write [

] as a linear combination of the vectors in this basis.

Problem 16 Show that { } is a basis for

. Also, write as a linear combination of the vectors from this basis.

Problem 17 Assume that be a nonempty finite set of vectors in a vector space If is a

vector in that can be written as a linear combination of other vectors in , and { } is that set

obtained by removing the vector , then show that span { } span( ) .

Problem 18 Suppose that be a nonempty finite set of vectors in a finite dimensional vector

space Use Problem 16 to show that if spans , but is not a basis for , then can be

reduced to a basis for by removing appropriate vectors from

Problem 19 Suppose that is an -dimensional vector space and is a subset of that

contains precisely -vectors. Prove that if span( ) = , then is a basis for .

Problem 20 Recall that { } and that is a vector

space. Show that the vector space is infinite-dimensional (Hint: Suppose it is finite

dimensional, then any set with more than that number of vectors would be dependent, but

consider sets of vectors with one in a particular component and all other components zero, first

component, second component, etc.)

*Problem 21 Prove that any subspace of a finite-dimensional vector space must also be a finite-

dimensional vector space (ask me if you need help, this is a tough one)

*Problem 22 Suppose that is the subspace of given by

{ } and is the subspace of given by

{ }. Then find a basis for and

Note: for this problem, you can assume that are subspaces of and don’t need to prove

this fact.

Problem 23 Give an example of an infinite dimensional vector space that contains a finite dimensional

subspace and describe the subspace including the finite basis for subspace.

Section 5 Row, Column and Null Spaces

Problem 1 Let [

] [

]. Express as a linear combination of the

column vectors of

Problem 2 Let [

] [

] Then determine whether is in the

column space of and if so write as a linear combination of the columns of .

Problem 3 In each part find the vector form of the general solution of the given linear system

and then use this result to find the vector form of the general solution of .

I

II

Problem 4 Find a basis for the null space of each matrix

I [

] II [

]

Problem 5 For each matrix find a basis for the row space of by reducing the matrix to

row echelon form.

I [

] II [

]

Problem 6 For each matrix find a basis for the row space of consisting of only row vectors

of

I [

] II [

]

Problem 7 Find a subset of the given set of vectors that forms a basis for the space spanned by

the given set of vectors.

I

II

Problem 8 Let be a subspace of Then the orthogonal complement of

{ }. Prove that is a subspace of and that { }.

*Problem 9 Prove that the orthogonal complement of is (one direction of this is difficult-

you will need to use the fact that every vector in can be expressed uniquely in the form

where comes from and comes from ).

Problem 10 Suppose that { } is a basis for of subspace of Show that the

orthogonal complement of consists of those vectors that are orthogonal to all of these basis

vectors.

Problem 11 Let matrix. Prove that and

i.e., the orthogonal

complement of the row space is the null space and the orthogonal complement of the null space

is the row space (Hint: Problem 10).

Problem 12 Find a basis for the orthogonal complement of subspace of spanned by the

given set of vectors. Note that the orthogonal complement of row space of a matrix is the null

space of that same matrix.

I { }

II { }

III { }

*Problem 13 Let matrix and suppose that { } is a basis for with

{ } as basis for Show that { } is a basis for (Hint:

the Dimension Theorem and Problem 8 may be helpful).

*Problem 14 Suppose that { } is a set of pairwise orthogonal vectors in

Show that ‖ ‖ ‖ ‖

‖ ‖ ‖ ‖

.

Note: this is a generalization of the Pythagorean Theorem.

Problem 15 Let invertible matrix. Prove that the row vectors of are form a

basis for

Problem 16 Show that if an matrix and rank( , then is invertible (Hint: use the

Dimension Theorem).

Problem 17 For each matrix find the number of leading variables and the number of

parameters in the solution of the homogeneous system .

I [

] II [

]

III [

]

Problem 18 Find the rank and nullity for each matrix from Problem 17 and verify that the

values satisfy the Dimension Theorem.

Chapter 5 Homework for Eigenvalues and Eigenvectors

Section 1 Eigenvalues and Eigenvectors

Problem 1 Which of the vectors [ ] [

] [

] [ ] are eigenvectors for the

matrix [

]. For any that are eigenvectors, find the corresponding eigenvalues.

Problem 2 Let such that [

] is an eigenvector corresponding to the eigenvalue 3

and [

] is an eigenvector corresponding to the eigenvalue 2. Now if [ ] , Compute .

Problem 3 Let such that [

] is an eigenvector corresponding to the eigenvalue 3

and [

] is an eigenvector corresponding to the eigenvalue 2. Now if [ ] , Compute .

Problem 4 Suppose an invertible Prove that is an eigenvalue of (Hint: the proof should be short).

Problem 5 Suppose that is an invertible matrix with eigenvalues n. Show that has

eigenvalues 1/n.

Problem 6 Suppose an Prove that is an eigenvalue of

.

Problem 7 Suppose that are two matrices and that is an eigenvector for with

corresponding eigenvalue 5 and is also an eigenvalue for but corresponding to the eigenvalue 7.

Show that 12 is an eigenvalue of

Problem 8 Find the characteristic polynomial for each matrix

I [

] II [

]

Problem 9 Let be a matrix and a particular eigenvalue of with and eigenvector

corresponding to . Prove that for any scalars c and k, c k is an eigenvalue for and is

an eigenvector corresponding to ck

Problem 10 Let be a matrix and a particular eigenvalue of . Show { = } is a

subspace of (note that this subspace is called the eigenspace corresponding to the eigenvalue ).

Problem 11 Find all eigenvalues and corresponding to each given matrix, and for each eigenvalue find a

basis for the eigenspace corresponding to that eigenvalue. Compare the geometric and algebraic

multiplicity for each eigenvalue.

I [

] II [

] III [

] IV [

]

Problem 12 Let be a matrix. Prove that is invertible if and only if zero is not an eigenvalue

of .

*Problem 13 Let be a matrix. Prove that is equal to the product of all of the

eigenvalues of counting algebraic multiplicities.

Problem 14 Suppose that and that the characteristic polynomial of

divides the characteristic polynomial of Prove that have exactly the same eigenvalues.

Problem 15 Let with an eigenvalue of Show that if p( x )=

then p( is an eigenvalue of .

Problem 16 Let with an eigenvalue of Show that if p(x) is any polynomial of

degree with real coefficients then p( is an eigenvalue of

Problem 17 Suppose that { for some 2x2 invertible matrix }.

Show that is not a vector space, by exhibiting a vector space axiom that fails to hold.

Section 2 Normal Forms and Diagonalization

Problem 1 Let [

] . Then find all eigenvalues of and a basis for each corresponding

eigenspace. Then find an invertible matrix

Problem 2 [

]. Then find all of the eigenvalues of and for each eigenvalue,

determine a basis for the eigenspace corresponding that eigenvalue. Also, determine if the matrix is

diagonalizable.

Problem 3 Determine whether each matrix is diagonalizable, and for those that are not diagonalizable

explain why we have too few linearly independent eigenvectors. For those matrices that are

diagonalizable find a matrix .

I [

] II [

] III [

]

IV [

] V [

]

Problem 4 Suppose that and that such

that Show that the matrices have

exactly the same eigenvalues.

*Problem 5 Let be a diagonalizable matrix. Prove that for any scalars the matrix

is also diagonalizable

Problem 6 Suppose that is a 3 matrix with three distinct eigenvalues and let { } be

three distinct eigenvectors corresponding to the eigenvalues. Show this set of eigenvectors must

be linearly independent.

Problem 7 Suppose that is a 3 matrix with three distinct eigenvalues. Show that is

diagonalizable.

Problem 8 Let be a diagonalizable matrix. Show that for any positive integer is

diagonalizable.

Problem 9 Let be a matrix that is not invertible. Prove that zero is an eigenvalue of and that

that is the eigenspace corresponding to the eigenvalue zero.

*Problem 10 Prove that similar matrices have the same rank.

Problem 11 Prove that similar matrices have the same nullity (Hint: The Dimension Theorem and Prob.

10).

Problem 12 Let be a matrix that is lower triangular. Prove that if the diagonal entries are all

distinct then is diagonalizable.

Problem 13 Let [

] and find a diagonal matrix that is similar to and use this to find .

Problem 14 Let be a matrix with real entries such that all of the columns and rows sum to a

constant . Show that is an eigenvalue of .

Chapter 6 Homework for Inner Product Spaces

Section 1 Basics of Inner Product Spaces

Problem 1 Compute ⟨ ⟩ using the standard inner polynomial inner product in .

I

II

Problem 2 Let ⟨ ⟩ , where [

] and . Then this defines an inner

product on called the matrix inner product. Let [ ] [

] [

] Calculate

each of the following:

I ⟨ ⟩ II ⟨ ⟩ III ⟨ ⟩ IV ⟨ ⟩

V ‖ ‖ ‖ ‖ VI ⟨ ⟩

Problem 3 For any [ ⁄ ] ⟨ ⟩ ∫ ⁄

. Then with this

inner product, [ ⁄ ] becomes an inner product space. Let and

in [ ⁄ ]. Calculate each of the following:

I ⟨ ⟩ II ⟨ ⟩ III ⟨ ⟩ IV ⟨ ⟩

V ‖ ‖ ‖ ‖

Problem 4 Suppose that are vectors in an inner product space with ⟨ ⟩

⟨ ⟩ , ⟨ ⟩ , ‖ ‖ ‖ ‖ , and ‖ ‖ . Find the value of each

expression.

I ⟨ ⟩ II ⟨ ⟩ III ⟨ ⟩ IV ‖ ‖

V ‖ ‖ VI ⟨ ⟩

Problem 5 Show that for any nonzero vector in an inner product space then ⟨

‖ ‖

‖ ‖⟩

Problem 6 Compute ‖ ‖ find the vector

‖ ‖ for each using the standard inner

polynomial inner product in .

I

II

Section 2 Orthogonality in Inner Product Spaces

Problem 1 Using the Euclidean inner product in find the cosine of the angle between

.

Problem 2 For find the cosine of the angle between

using the standard inner polynomial inner product in .

Problem 3 Let be two vectors in the inner product space [ ] with

integral inner product. Then find the angle between these two vectors.

Problem 4 Let { } and let Find a basis for the orthogonal

complement of (Hint: we already did this back in Chapter 4)

Problem 5 Let { } and let Find a basis for

the orthogonal complement of

Problem 6 Show that if { } is an orthogonal set of nonzero vectors in an inner

product space then is linearly independent.

Problem 7 Let be vectors in an inner product space such that is orthogonal to

the vectors Use properties of inner products to show that for any constants

is orthogonal to .

Problem 8 In consider the subspace { }. Then find a basis for

the orthogonal complement of using the Euclidean inner product and write the vector

as a sum of two vectors in where one vector is in and the other is in the

orthogonal complement of .

*Problem 9 For [ ] we use the inner product ⟨ ⟩ ∫

. Then

show that { √ √ ⁄⁄ } is an orthonormal set and that is an orthonormal basis for

Then for , express as a sum of two vectors where one vector is in

and the other is in the orthogonal complement of using this inner product.

Problem 10 Suppose that { } is a basis for of subspace of an inner product space

Show that the orthogonal complement of consists of those vectors that are orthogonal to all

of these basis vectors using the inner product for the vector space.

Problem 11 Let be a subspace of inner product space Then the orthogonal complement of

{ ⟨ ⟩ }. Prove that is a subspace of and that

{ }.

*Problem 12 Let be a subspace of an inner product space Show that the orthogonal

complement of the orthogonal complement of is . (Hint: Use the fact that every vector in

can be written as a sum of two vectors, where one vector is and the other is in the orthogonal

complement of )

Problem 13 Let be a subspace of a finite dimensional inner product space and let

{ } be a basis for with { } as basis for the orthogonal complement of

Show that { } is a basis for

Problem 14 Suppose that is a subspace of and let { } be a basis for the orthogonal

complement of Assuming that , find a basis for

Section 3 Gram-Schmidt

Problem 1 Erhardt Schmidt died in 1959 and Jorgen Gram died in 1916. At the time of death,

which man was older? (Hint: Google)

Problem 2 Show that { √ ⁄ √ ⁄ √ ⁄ √ ⁄ } is an orthonormal basis

for using the Euclidean inner product.

Problem 3 Show that { ⁄ ⁄ ⁄ ⁄ } is an orthonormal basis for

using the Euclidean inner product at and use the Euclidean inner product to write as

a linear combination of the vectors in .

Problem 4 Using several complete sentences, describe some of the advantages of an

orthonormal basis as compared with a basis that is not orthonormal.

Problem 5 Suppose that and let be the subspace of spanned by vectors

. Then fine .

Problem 6 Let [

] . Then has only one eigenvalue 2. Find a basis for the eigenspace

and find the projection of the vector [

] on the eigenspace i.e., find the vector

Problem 7 Suppose that { } is an orthogonal basis for and is a nonzero vector in

. Prove that

‖ ‖

‖ ‖

‖ ‖ .

Problem 8 Use the Gram-Schmidt method to construct an orthonormal basis for using each

given basis: I { } II { }

Problem 9 The set { } is a basis for . Then using this basis, the

Euclidean inner product and the Gram-Schmidt method construct an orthonormal basis for

Problem 10 Use the Gram-Schmidt method to find an orthogonal basis for the span of

{ } in using the Euclidean inner product.

Problem 11 Use the Gram-Schmidt method to construct an orthonormal basis for using the

basis { }

Problem 12 Use the Gram-Schmidt method to construct an orthonormal basis for the subspace

of given by the hyperplane { }.

Problem 13 Consider the vector space of all polynomials in the variable with degree less

than or equal to 2, which has a standard basis { }. Then for we

use the inner product ⟨ ⟩ ∫

. Use the Gram-Schmidt method to construct an

orthonormal basis for . Note the elements of this orthonormal basis are known as the first three

Legendre polynomials.

Problem 14 Using the inner product of Problem 12, write as a linear

combination of the first three Legendre polynomials.

Problem 15 Consider the vector space of all polynomials in the variable with degree less

than or equal to 3, which has a standard basis { }. Then for

we use the inner product ⟨ ⟩ ∫

. Use the Gram-Schmidt method to construct

an orthonormal basis for .

Problem 16 Suppose that is a subspace of and let { } be a basis for the orthogonal

complement of Assuming that , find an orthonormal basis for

Problem 17 In the Euclidean inner product space find the projection of the vector

on the space determined by the equation

Section 4 Orthogonal Diagonalization

Problem 1 Let [

] . Then find an orthogonal matrix such that

is a diagonal matrix and determine (Hint: use Gram-Schmidt on the linearly

independent eigenvectors from the eigenspaces)

Problem 2 Let [

] . Then find an orthogonal matrix such

that

is a diagonal matrix and determine .

Problem 3 Let be as in Problem 1 above and let { } be the successive column vectors of

your orthogonal matrix found in Problem 1. Then if are eigenvalues corresponding respectively

to then we say that

is the Spectral Decomposition of Find the

spectral decomposition of .

Problem 4 Let be as in Problem 2 above and let { } be the successive column vectors of

your orthogonal matrix found in Problem 2. Then if are eigenvalues corresponding

respectively to then we say that

is the Spectral

Decomposition of Find the spectral decomposition of .

Problem 5 If { } is an orthogonal basis for and

then

prove that is a symmetric matrix.

Section 5 Best Approximation and Fourier Series (if time)

Problem 1 Find the least square quadratic polynomial approximation for on [ ].

Problem 2 Find the least square linear polynomial approximation for on [ ].

Problem 3 Find a least squares approximation for over the interval [ ] by

(a) a trig polynomial of order 2 or less

(b) a trig polynomial of order n or less

Problem 4 Find a least squares approximation for over the interval [ ] by

(a) a trig polynomial of order 3 or less

(b) a trig polynomial of order n or less

Chapter 7 Homework Linear Transformations

Section 1 General Linear Transformations

Problem 1 Consider the basis { } where Let be the

linear operator such that Then find a formula for

and use this formula to find .

Problem 2 Consider the basis { } where and

Let be the linear operator such that and

Then find a formula for and use this formula to find .

Problem 3 Let be the linear transformation such that

and . Find .

Problem 4 Let . Show directly that the is a subspace of and that

. Show that is a subspace of and that .

Problem 5 Let be given by

. Determine

if

Problem 6 Suppose [ ] [ ] is given by . Show that is a linear

transformation.

Problem 7 Suppose that [ ⁄ ] [ ⁄ ] is given by . Show

that is a linear transformation and find for Is

Problem 8 Suppose [ ] [ ] is given by the derivative of the function

Find the and a basis for the kernel.

Problem 9 Suppose [ ] [ ] is given by for all in the

interval [ ] is the derivative of the function Find the and a basis for the

kernel.

Problem 10 Consider the linear transformation which is given

by The kernel of this linear transformation is a two dimensional subspace of

. Given the fact that the kernel of has dimension two, find a basis for the kernel

and show that your basis is a basis.

Problem 11 Show that given by ( ) is a linear transformation.

Problem 12 Suppose [ ] [ ] is given by for all in the

interval [ ] Show that is a linear transformation and find

Problem 13 Let be a linear transformation from a vector space to a vector space

. Prove that ( ) .

Problem 14 Let be given by . Show that is a linear

transformation. Also, find

Problem 15 Let be a linear operator on a finite dimensional vector space with

basis { }. Prove that if then is the

identity transformation on

Problem 16 Suppose is given by

Show that is a linear transformation and that and are in Then

assuming the dimension of find a basis for

Section 2 Isomorphism

Problem 1 Let be given by Show that is an

isomorphism.

Problem 2 Suppose that is an inner product space with dimension 3 and is a given fixed

nonzero vector in . We then define by ⟨ ⟩. Show that is a linear

transformation. Is an isomorphism?

Problem 3 Let be a linear transformation from the vector space to the vector space

such that { }. Show that is 1-1.

Problem 4 Let be a linear transformation from the vector space to the vector space

such that is 1-1. Show that { }.

Problem 5 Let be a linear operator, where is a finite dimensional vector space.

Show that if { } then is onto and so .

*Problem 6 Show that Problem 5 is not true if we drop the finite dimensional condition i.e.,

find a vector space and a linear transformation such that { } , but

.

Problem 7 Suppose that is a invertible matrix. Show that given by

is an isomorphism.

Problem 8 Suppose that is a matrix that is not invertible. Show that given

by is a linear operator, but not an isomorphism.

Problem 9 Suppose that is the isomorphism given by

where

is the rotation matrix [

] . Let be a nontrivial, proper subspace of with

orthogonal complement using the Euclidean inner product. Show (

)

Problem 10 Give an example of an isomorphism and a nontrivial, proper subspace

of with orthogonal complement such that ( ) using the Euclidean

inner product.

*Problem 11 Suppose that is the subspace of determined by

. Find the orthogonal complement of . Also, if [

] , which is

the matrix and that rotates a vector in about the , let is given by the

linear matrix transformation . Show that

( )

Problem 11 Let be an isomorphism between a finite dimensional vector space and

a finite dimensional vector such that { } is a basis for . Show that

{ } is a basis for .

Problem 12 Suppose that are both vector spaces with dimension . Prove that

are isomorphic.

*Problem 13 Suppose that are isomorphic finite dimensional vector spaces. Prove

that they have the same dimension.

Problem 14 Find the standard matrix for the linear operator given by

and show that is both 1-1 and onto.

Problem 15 If is a linear transformation from a vector space to a vector space then

{ }

(i) Prove the is a subspace of

(ii) Prove that if the contains only the zero vector of 1-1.

Problem 16 Suppose that is an isomorphism form a finite dimensional vector space

to a vector space and { } is a basis for Prove that

{ } is a linearly independent set in Note you can’t just state the result

from the text or notes, you need to prove this directly.

Problem 17 Suppose that and that the linear

transformation is given by

(i) Show that the linear transformation is not one to one.

(ii) Show that the linear transformation is not onto.

Problem 18 Consider the linear transformation where given by

. Then find the matrix for this linear transformation [ ] and all

eigenvalues and the corresponding eigenspace for each eigenvalue.

Problem 19 Consider the linear transformation where given by

. Then find the matrix for this linear transformation [ ] and all

eigenvalues and the corresponding eigenspace for each eigenvalue.

Section 3 Linear Transformations Involving

Problem 1 Find the standard matrix for the transformation determined by the system

Problem 2 Find the standard matrix for the linear operator defined by the given formula.

I

II

Problem 3 Use matrix multiplication to find the reflection of (-2,3) about the line

Problem 4 Use matrix multiplication to find the reflection of (2, -4, 3) about the plane.

Problem 5 Use matrix multiplication to find the orthogonal projection of (2,-1,3) on the

plane.

Problem 6 Use matrix multiplication to find the image of (2, -4) when it is rotated through an

angle of I and II .

Problem 7 Use matrix multiplication to find the image of the vector when it is rotated

about the

Problem 8 Let and . Find the

standard matrices for and and use these to find the standard matrices for and

Problem 9 Suppose we have two linear transformations and with

and . Find the

standard matrix for .

Problem 10 Find the standard matrix for each stated composition.

I A rotation of counterclockwise, followed by a reflection about .

II An orthogonal projection on the followed by a dilation by a factor of 2.

III A clockwise rotation followed by a reflection about , and the a reflection about

the .

Coordinate Vectors

If { } is a basis for an dimensional vector space and

is a vector in then [ ] is called the coordinate vector of with respect to

the basis and the mapping [ ] defines an isomorphism.

Problem 11 Find [ ] when using the basis { }

for .

Problem 12 Find [ ] when in , where in this particular case we have

{ } for our basis in .

Problem 13 Suppose that [

] [

] [

] and also that

[

] [

] [

] . Then { } and { }

are both bases for . Find a such that , and

.

Problem 14 Suppose that [ ] [

] and also that

[

] [ ] . Then { } and { }

are both bases for . Find a such that [ ] [ ] , for all

*Problem 15 Suppose is a basis for . Show that if are vectors in

then { } is a linearly independent set in if and only if the set

{[ ] [ ] [ ] } is a linearly independent set in . Hint: We can define a linear

operator from by [ ] .

Problem 16 Find the standard matrix for the linear operator given by

and show that is both 1-1 and onto.

Problem 17 Suppose that and is given by for all vectors

in Show that if .

Problem 18 Find the projection of the vector on the line that passes through the origin and

makes a 60 degree angle with the positive

Problem 19 Let the linear operator that reflects each vector about the line and then

rotates the resulting vector 90 degrees counterclockwise. Then let be the linear operator that

dilates a vector by a factor of 3 and then reflects the resulting vector about the Then

find the vector in .

Problem 20 Let the linear operator that reflects each vector about the line and

then rotates the resulting vector 45 degrees counterclockwise. Then let be the linear operator

that dilates a vector by a factor of 3 and then reflects the resulting vector about the Then

find the vector in .

Problem 21 Let { }. Then we will consider that two

linear transformations are the same if Show that is a vector

space, where

Problem 22 Let { } as in the above. Show that has

finite dimension (Hint: each linear transformation has a matrix representation in this case).

Problem 23 Let be a vector space and { }, where addition and

scalar multiplication is defined by

I for all and

II for all and for all scalars

Show that the set is not a vector space by exhibiting a vector space axiom that fails to hold.

Problem 24 Let C be the set of all complex numbers. Then show that C is a real vector space. If we

define C by show that is an isomorphism so that C are isomorphic

vector spaces.