Unit 4: Vector Spaces and Linear Transformations *This unit discusses vector spaces and linear transformations. The first topic you will study in this unit is that of an abstract vector space. You will see that a vector space is a collection of vectors that satisfies a set of axioms. Next, you will study ideas such as subspaces, linear independence, and bases in the context of vector spaces. Remember to think of R or C if you are confused. Next, you will construct abstract fields and vector spaces. You will begin this by first reviewing some basic algebra relating to polynomials. This is both interesting and important, because it provides the basis for constructing different kinds of fields. Finally, you will look at inner product spaces, which are vector spaces that also have an inner product, before moving on to linear transformations, which is the second topic of study in this unit.
Linear transformations have many applications within mathematics as well as in fields outside of mathematics, such as physics. You will study the basic definitions of linear transformations and properties and relations between these and matrices. *
Unit 4 Time Advisory
This unit should take you approximately 29.5 hours to complete.
☐ Subunit 4.1: 19 hours ☐ Sub-subunit 4.1.1: 4 hours
☐ Reading: 2 hours
☐ Assignment: 2 hours
☐ Sub-subunit 4.1.2: 7.5 hours
☐ Reading: 3.5 hours
☐ Assignment: 4 hours
☐ Sub-subunit 4.1.3: 7.5 hours
☐ Reading: 3.5 hours
☐ Assignment: 4 hours
☐ Subunit 4.2: 10.5 hours ☐ Sub-subunit 4.2.1: 0.5 hour
☐ Sub-subunit 4.2.2: 1.5 hours
☐ Sub-subunit 4.2.3: 1 hour
☐ Sub-subunit 4.2.4: 2 hours
☐ Sub-subunit 4.2.5: 5.5 hours
☐ Reading: 1.5 hours
☐ Assignment: 4 hours
Unit4 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- define vector spaces;
- state the axioms of a vector space;
- prove the Euclidean algorithm;
- define linear independence;
- determine if a given subset of a vector space is a subspace;
- determine if a set of vectors is a basis for a vector space;
- find the dimension of a subspace;
- state and prove the exchange theorem;
- define and provide examples of inner product spaces;
- state and prove the Cauchy-Schwarz inequality;
- apply the Gram-Schmidt process to inner product spaces;
- determine if a transformation is linear;
- determine if a function from one vector space to another is a linear
transformation;
- find matrix representations for a given linear transformation;
- find the range and kernel of a transformation;
- use linear transformations to prove that vector spaces are
isomorphic;
- explain which matrices are diagonalizable;
- determine if given sets of vectors are orthogonal and find
orthogonal projections;
- solve least squares problems; and
- find the Fourier approximation for a known function.
4.1 Vector Spaces
4.1.1 Algebraic Considerations, Linear Independence, and Bases
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 16: Vector Spaces:” “Section 16.1: Algebraic
Considerations” and “Section 16.2.1: Linear Independence and
Bases”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
16: Vector Spaces:” “Section 16.1: Algebraic Considerations” and
“Section 16.2.1: Linear Independence and Bases” (PDF)
Instructions: Please click on the link above, and read Section 16.1
on page 323 and section 16.2.1 on pages 325–330. Section 16.1 will
provide the definition of a vector space, and Section 16.2.1 will
discuss subspaces, linear dependence, and bases. These readings
should take you approximately 2 hours to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 16: Vector Spaces:” “Section 16.2: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
16: Vector Spaces:” “Section 16.2:
Exercises”
(PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 330, and complete problems 1, 2, 3, and 4. Next, click on “Solutions” (PDF) and check your answers on pages 124–125. This assessment should take you approximately 2 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
4.1.2 Vector Spaces and Fields
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 16: Vector Spaces:” “Section 16.3: Vector Space and
Fields”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
16: Vector Spaces:” “Section 16.3: Vector Space and Fields” (PDF)
Instructions: Please click on the link above, and read Section 16.3
on pages 330–343. Section 16.3 will discuss vector space and
fields. This reading should take you approximately 3 hours and 30
minutes to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 16: Vector Spaces:” “Section 16.4: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
16: Vector Spaces:” “Section 16.4: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 243, and work on problems 2, 4, 10, 12, 16, 20, 24, and 29. Next, click on “Solutions” (PDF) and check your answers on pages 125–135. This assessment should take you approximately 4 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
4.1.3 Inner Product Spaces
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 16: Vector Spaces:” “Section 16.5: Inner Product Spaces”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
16: Vector Spaces:” “Section 16.5: Inner Product Spaces” (PDF)
Instructions: Please click on the link above, and read Section 16.5
on pages 348–361. Section 16.5 will discuss vector spaces with
inner products. This reading should take you approximately 3 hours
and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 16: Vector Spaces:” “Section 16.6: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
16: Vector Spaces:” “Section 16.6: Exercises” (PDF)
Instructions: Please click on the above link to open the PDF. Scroll down to page 361, and complete problems 1, 4, 7, 12, 15, 17, and 22. Next, click on “Solutions” (PDF) and check your answers on pages 135–150. This assessment should take you approximately 4 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
4.2 Linear Transformations
4.2.1 Matrix Multiplication and L(V,W)
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 17: Linear Transformations:” “Section 17.1: Matrix
Multiplication as a Linear Transformation” and “Section 17.2: L(V,W)
as a Vector Space”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra:“Chapter
17: Linear Transformations:” “Section 17.1: Matrix Multiplication as
a Linear Transformation” and “Section 17.2: L(V,W) as a Vector
Space” (PDF)
Instructions: Please click on the link above, and read the
indicated sections on pages 367 and 368. Sections 17.1 and 17.2
will discuss linear transformations and vector spaces. These
readings should take you approximately 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
4.2.2 Eigenvalues And Eigenvectors Of Linear Transformations
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 17: Linear Transformations:” “Section 17.3: Eigenvalues and
Eigenvectors of Linear Transformations”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
17: Linear Transformations:” “Section 17.3: Eigenvalues and
Eigenvectors of Linear Transformations” (PDF)
Instructions: Please click on the above link, and read Section 17.3
on pages 369–373. Section 17.3 will discuss finding eigenvalues and
eigenvectors of linear transformations. This reading should take
you approximately 1 hour and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
4.2.3 The Block Diagonal Matrices
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 17: Linear Transformations:” “Section 17.4: Block Diagonal
Matrices”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
17: Linear Transformations:” “Section 17.4: Block Diagonal Matrices”
(PDF)
Instructions: Please click on the link above, and read Section 17.4
on pages 373–377. Sections 17.4 will discuss linear
transformations, which will result by multiplication by n × n
matrices. This reading should take you approximately 1 hour to
complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
4.2.4 The Matrix of a Linear Transformation
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 17: Linear Transformations:” “Section 17.5: The Matrix of a
Linear Transformation”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
17: Linear Transformations:” “Section 17.5: The Matrix of a Linear
Transformatio” (PDF)
Instructions: Please click on the link above, and read Section 17.5
on pages 377–386. Section 17.4 will discuss matrices of linear
transformations. This reading should take you approximately 2 hours
to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
4.2.5 The Matrix Exponential, Differential Equations
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 17: Linear Transformations:” “Section 17.6: The Matrix
Exponential, Differential Equations”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
17: Linear Transformations:” “Section 17.6: The Matrix Exponential,
Differential Equations” (PDF)
Instructions: Please click on the link above, and read Section 17.6
on pages 386–391. Section 17.6 will introduce fundamental matrices,
matrices whose entries are differentiable functions. This reading
should take you approximately 1 hour and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra:
“Chapter 17: Linear Transformations:” “Section 17.7: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear
Algebra: “Chapter
17: Linear Transformations:” “Section 17.7: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 391, and work on problems 1, 3, 8, 12, 15, 18, and 21. Next, click on “Solutions” (PDF) and check your answers on pages 151–157. This assessment should take you approximately 4 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.