Unit 4: Inner Product Spaces, Self-adjoint Operators and the Spectral Theorem for Normal Maps Linear algebra deals with not only Euclidean spaces but also abstract vector spaces. This unit will discuss lengths and angles in an abstract vector space. Inner products allow us to generalize notions such as length, because an inner product is a generalization of a dot product for Euclidean n-space. Having notions of length, angles, and distances in an abstract vector space allow us to apply more tools and methods which help us to better understand the structure of the space.
In this unit, we will discuss inner product spaces, which are vector
spaces with an additional structure known as an inner product. Much of
the motivation for the subject grew from the need to generalize some
geometric properties of two-dimensional and three-dimensional Euclidean
spaces to higher dimensional spaces. In this unit, we will finally
begin to understand the geometric aspects of linear algebra, such as
representing rotations in the three-dimensional Euclidean space as
matrices. From this we will understand how to generalize and represent
rotations in higher dimensional Euclidean spaces as matrices. The
concepts in this unit, such as norm and inner product, provide structure
on spaces. We will finally study the basic properties of inner product
spaces, orthonormal bases, and the Gram-Schmidt orthogonalization
procedure. We will further study range-nullspace decomposition,
orthogonal decomposition, and singular-value decomposition of spaces.*
*Next, we will try to understand and answer the question of when a
linear operator on an inner product space is diagonalizable. We will
study the notion of an adjoint of an operator as well as normal
operators and then discuss the spectral theorem, which characterizes the
linear operators for which an orthonormal basis consisting of
eigenvectors exists. The spectral theory studied here is closely
related to that studied in Unit 2. In fact, the eigenvalues and
eigenvectors for a matrix are the same as those for the linear
transformation determined by the matrix. We will then learn about
finding the singular-value decomposition of an operator. We will
conclude by exploring some advanced topics.
Unit 4 Time Advisory
This unit will take you approximately 31.5 hours to complete.
☐ Subunit 4.1: 15.5 hours ☐ Sub-subunit 4.1.1: 3 hours
☐ Sub-subunit 4.1.2: 3 hours
☐ Sub-subunit 4.1.3: 9.5 hours
☐ Subunit 4.2: 16 hours ☐ Sub-subunit 4.2.1: 3 hours
☐ Sub-subunit 4.2.2: 3 hours
☐ Sub-subunit 4.2.3: 3 hours
☐ Sub-subunit 4.2.4: 7 hours
Unit4 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Define a normed vector space.
- Find an orthonormal basis for a given inner product space.
- Apply the Cauchy Schwarz inequality.
- Compute the tensor product of two vectors.
- State the Riesz representation theorem.
- Find the volume of a parallelepiped determined by three given
vectors.
- State what it means for a nxn matrix to be diagonalizable.
- Define and identify Hermitian operators.
- Define and identify Hilbert spaces.
- Prove the Cayley Hamilton theorem.
- Define and compute the adjoint of an operator.
- Define and identify normal operators.
- State the spectral theorem.
- Determine the singular-value decomposition of an operator.
- Define the notion of length for abstract vectors in abstract vector
spaces.
- Define and identify orthogonal vectors.
- Define and identify orthogonal and orthonormal subsets of R^{n.}
- Use the Gram-Schmidt process.
- Compute the range-nullspace decomposition given by a linear
transformation.
- Explain how the matrix for a linear operator changes if we change
from one orthonormal basis to another.
4.1 Inner Product Spaces 4.1.1 General Theory - Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces”
Link: Brigham Young University: Kenneth Kuttler’s *An Introduction
to Linear Algebra:* [“Chapter 12: Inner Product
Spaces”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/02/Linear-Algebra-Kuttler-1-30-11-OTC.pdf)
(PDF)
Instructions: Please read Section 12.1 (pages 287–289) in its
entirety. Note that the notion of an inner product space is a
generalization of the situation with Euclidean spaces.
Studying this reading should take approximately 1.5 hours to
complete.
Terms of Use: *An Introduction to Linear Algebra* was written by
Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the
Saylor Foundation’s Open Textbook Challenge.
Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 9: Inner Product Spaces”
Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 9: Inner Product Spaces” (PDF)
Instructions: Please read Sections 9.1–9.4. Note the parallels to the situation with Euclidean spaces.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: . These materials have been reproduced for educational and non-commercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
4.1.2 The Gram-Schmidt Process - Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 9: Inner Product Spaces”
Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne
Schilling’s *Linear Algebra: As an Introduction to Abstract
Mathematics*: [“Chapter 9: Inner Product
Spaces”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/10/mat67_course_notes.pdf)
(PDF)
Instructions: Please read Section 9.5. The Gram-Schmidt process can
be used to turn any basis for an inner product space into an
orthogonal basis.
Studying this reading should take approximately 1.5 hours to
complete.
Terms of Use: These materials have been reproduced for educational
and non-commercial purposes and can be viewed in their original
format
[here](https://mail.whittier.edu/owa/redir.aspx?C=fa77263df66842f9a74d8d1b871cd15b&URL=http%3A%2F%2Fwww.math.ucdavis.edu%2F~anne%2Flinear_algebra%2Fmat67_course_notes.pdf).
Any reproduction or redistribution for commercial use is strictly
prohibited.
Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces”
Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces” (PDF)
Instructions: Please read Section 12.2 (pages 289–292) in its entirety. The Gram-Schmidt process can be used to turn any basis for an inner product space into an orthogonal basis.
Studying this reading should take approximately 1.5 hours to complete.Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
4.1.3 Advanced Topics - Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 9”
Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne
Schilling’s *Linear Algebra: As an Introduction to Abstract
Mathematics*: [“Exercises for Chapter
9”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/10/mat67_course_notes.pdf)
(PDF)
Instructions: Please complete calculational exercises 1, 3, and 5
and proof-writing exercises 1, 2, 5, and 6 (pages 133–135).
Completing this activity should take approximately 2.5 hours to
complete.
Terms of Use: These materials have been reproduced for educational
and non-commercial purposes and can be viewed in their original
format
[here](https://mail.whittier.edu/owa/redir.aspx?C=fa77263df66842f9a74d8d1b871cd15b&URL=http%3A%2F%2Fwww.math.ucdavis.edu%2F~anne%2Flinear_algebra%2Fmat67_course_notes.pdf).
Any reproduction or redistribution for commercial use is strictly
prohibited.
Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 9: Inner Product Spaces”
Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 9: Inner Product Spaces” (PDF)
Instructions: Please read Section 9.6 on pages 128–132. Work through the examples on your own, and compare your work with that in the text.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: These materials have been reproduced for educational and non-commercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces”
Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces” (PDF)
Instructions: Please read Sections 12.3–12.8 (pages 292–305) in their entirety. If the proofs are difficult to understand, try working through them in the low dimensional setting first.
Studying this reading should take approximately 1.5 hours to complete.Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces”
Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces” (PDF)
Instructions: Please work through problems 1–3, 9, 11, 14, 16, 21, and 22 in Section 12.7 (pages 299–302) and problems 1 and 3 in Section 12.9 (page 306). When you are done, check your solutions with the answers on pages 495 and 496.
This activity should take approximately 4.5 hours to complete.Terms of UseAn Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
4.2 Spectral Theorem 4.2.1 Self Adjoint Operators - Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps”
Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne
Schilling’s *Linear Algebra: As an Introduction to Abstract
Mathematics*: [“Chapter 11: The Spectral Theorem for Normal Linear
Maps”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/10/mat67_course_notes.pdf)
(PDF)
Instructions: Please read Sections 11.1 and 11.2. Work through the
examples on your own, and compare your work with that in the text.
Studying this reading should take approximately 1.5 hours to
complete.
Terms of Use: These materials have been reproduced for educational
and non-commercial purposes and can be viewed in their original
format
[here](https://mail.whittier.edu/owa/redir.aspx?C=fa77263df66842f9a74d8d1b871cd15b&URL=http%3A%2F%2Fwww.math.ucdavis.edu%2F~anne%2Flinear_algebra%2Fmat67_course_notes.pdf).
Any reproduction or redistribution for commercial use is strictly
prohibited.
Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators”
Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators” (PDF)
Instructions: Please read Sections 13.1 and 13.2 (pages 307–312) in their entirety. Note that Schur's Theorem reappears in this generalized setting.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
4.2.2 Spectral Theorem - Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps”
Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne
Schilling’s *Linear Algebra: As an Introduction to Abstract
Mathematics*: [“Chapter 11: The Spectral Theorem for Normal Linear
Maps”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/10/mat67_course_notes.pdf)
(PDF)
Instructions: Please read Sections 11.3 and 11.4. The Spectral
Theorem describes the relationship between normal operators and
eigenvectors. Work through the examples on your own, and compare
your work with that in the text.
Studying this reading should take approximately 1.5 hours to
complete.
Terms of Use: These materials have been reproduced for educational
and non-commercial purposes and can be viewed in their original
format
[here](https://mail.whittier.edu/owa/redir.aspx?C=fa77263df66842f9a74d8d1b871cd15b&URL=http%3A%2F%2Fwww.math.ucdavis.edu%2F~anne%2Flinear_algebra%2Fmat67_course_notes.pdf). Any
reproduction or redistribution for commercial use is strictly
prohibited.
Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators”
Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators” (PDF)
Instructions: Please read Section 13.3 (pages 312–316) in its entirety. Here, you will learn about Hilbert spaces and their properties.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
4.2.3 Positive and Negative Operators - Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators”
Link: Brigham Young University: Kenneth Kuttler’s *An Introduction
to Linear Algebra:* [“Chapter 13: Self Adjoint
Operators”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/02/Linear-Algebra-Kuttler-1-30-11-OTC.pdf)
(PDF)
Instructions: Please read Sections 13.4 and 13.5 (pages 317–321) in
their entirety. You will read about the possibility of taking
fractional powers of certain linear operators.
Studying this reading should take approximately 1.5 hours to
complete.
Terms of Use: *An Introduction to Linear Algebra* was written by
Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the
Saylor Foundation’s Open Textbook Challenge.
Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps”
Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps” (PDF)
Instructions: Please read Section 11.5. Work through the examples on your own, and compare your work with that in the text.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: These materials have been reproduced for educational and non-commercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
4.2.4 Decomposition and Applications - Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps”
Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne
Schilling’s *Linear Algebra: As an Introduction to Abstract
Mathematics*: [“Chapter 11: The Spectral Theorem for Normal Linear
Maps”](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/10/mat67_course_notes.pdf)
(PDF)
Instructions: Please read Sections 11.6 and 11.7. The
singular-value decomposition generalizes the notion of
diagonalization.
Studying this reading should take approximately 1 hour to
complete.
Terms of Use: These materials have been reproduced for educational
and non-commercial purposes and can be viewed in their original
format
[here](https://mail.whittier.edu/owa/redir.aspx?C=fa77263df66842f9a74d8d1b871cd15b&URL=http%3A%2F%2Fwww.math.ucdavis.edu%2F~anne%2Flinear_algebra%2Fmat67_course_notes.pdf).
Any reproduction or redistribution for commercial use is strictly
prohibited.
Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 11”
Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 11” (PDF)
Instructions: Please complete calculational exercises 1 and 6 and proof-writing exercises 3 and 4 (pages 158–160).
Completing this activity should take approximately 3 hours to complete.
Terms of Use: These materials have been reproduced for educational and non-commercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators”
Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators” (PDF)
Instructions: Please read Sections 13.6–13.11 (pages 322–334) in their entirety. Here, you will learn about the singular-value decomposition of a matrix, which has applications in statistics and image analysis.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations”
Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations” (PDF)
Instructions: Please work through problems 13, 15, 16, and 19 in Section 13.12. When you are done, check your solutions with the answers on page 496.
This activity should take approximately 2 hours to complete.Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CC-BY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.