# MA232: Abstract Algebra II

Unit 3: Vector Spaces   Vector spaces are among the most useful structures in mathematics.  Used heavily in economics and finance as well as engineering and the natural and physical sciences, vector spaces are additional structures that have both algebraic and geometric properties.

Vectors are extended commutative groups with additional distributive properties concerning field values called scalars.  Thus, all theorems that apply to groups may apply to vectors.

A *
subspace** is a subset of a vector space that contains the zero vector and is a vector space itself.  Linear independence is a property of spaces and subspaces that states that no family of vectors in the space may be written as linear combinations of the other vectors in the family.  The number of unique, linearly independent vectors in a space is called the space’s dimension.*

The *
basis** of a space is a set of vectors that can represent all the vectors in the space by linear combinations.  That is, the basis is a linearly independent spanning set of the space.  Sometimes, it is easier to work in some bases than others.  For that reason, we sometimes prefer to change a basis from one coordinate set to another.  The group isomorphism that maps one basis to another is called a change of basis.  This type of isomorphism is a category of a set of functions called linear transformations.  In general, linear transformations are functions that preserve the operations of vector addition and scalar multiplication.  We will discover that all compositions of transformations result in transformations.  The kernel of a linear transformation L is the set of all vectors v in a space V for which L(v) = 0.  That is, all of the vectors that are mapped to the zero vector by L are in L’s kernel.  The kernel of L is by nature a subspace of the vector space V.  If the only vector in V contained in the kernel of L (also called Ker(L)) is 0, then L is 1-1.  The range of a transformation L is all vectors w in space W for which there is a v in space V such that L(v) = w.  If L is onto W, then the range of L = W.  If Ker(L) = {0} and range L = W, then L is a vector space isomorphism.  If L is an isomorphism, then matrices made from vectors in V are invertible.*

At the end of the unit, we will consider the Fundamental Theorem of Invertible Matrices, which is the core theorem of linear algebra.  The beauty of this theorem is that there are twenty equivalent statements about matrices.  If we determine that any of the twenty are true about a matrix, they are all true.  Conversely, if any is not true, none are true.

This unit will take approximately 45 hours to complete.

☐    Subunit 3.1: 4 hours

☐    Subunit 3.2: 4 hours

☐    Subunit 3.3: 5 hours

☐    Subunit 3.4: 5 hours

☐    Subunit 3.5: 6 hours

☐    Video: 1 hour

☐    Assignment: 0.5 hours

☐    Subunit 3.6: 5 hours

☐    Subunit 3.7: 6 hours

☐    Video: 0.5 hours

☐    Assignment: 1 hour

☐    Subunit 3.8: 5 hours

☐    Subunit 3.9: 5 hours

Unit3 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

- Determine whether a vector space is independent or not. - Find the basis of a vector space. - Use linear transformations to change bases. - Define the kernel of a linear transformation. - Determine if a matrix is invertible.

3.1 Definitions and Examples of Vector Spaces   - Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Vector Spaces” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Vector Spaces”

`````` Also available in:

– 321.

use displayed on pages 410 – 417.  The material linked above is
licensed under the [GNU Free Documentation
attributed to Thomas W. Judson and the original version can be
``````

Instructions: Please watch the entire lecture, which provides specific examples of vectors on a coordinate plane.

Watching this lecture should take approximately 30 minutes.

• Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Vector Spaces”: “Exercise Problems 3 and 5” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Vector Spaces”: “Exercise Problems 3 and 5”

Also available in:

iBooks

Instructions: Do problems 3 and 5 on page 325.  The solution can be found on page 407.

3.2 Subspaces   - Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Vector Spaces” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Vector Spaces”

`````` Also available in:

use displayed on pages 410 – 417.  The material linked above is
licensed under the [GNU Free Documentation
attributed to Thomas W. Judson and the original version can be
``````

Watching this lecture should take approximately 30 minutes.

• Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Vector Spaces”: “Exercise Problem 7” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Vector Spaces”: “Exercise Problem 7”

Also available in:

iBooks

Instructions: Do problem 7 on page 325.  The solution can be found on page 407.

3.3 Linear Independence   - Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Vector Spaces”: “Linear Independence” Link: St. Michael's College: Jim Hefferon’s “Linear Algebra” (PDF): “Vector Spaces”: “Linear Independence”

`````` Also available in:

99 – 106.

Notes on the Textbook: This PDF file will be used for the rest of
the unit.  It will be referenced for readings and assignments
throughout.

use displayed on pages iv – vi.  The material linked above is
licensed under the [GNU Free Documentation
attributed to Jim Hefferon and the original version can be
found [here](http://joshua.smcvt.edu/linearalgebra/) (HTML).
``````
• Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Vector Spaces” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Vector Spaces”

Also available in:

iBooks

1.

Watching this lecture should take approximately 15 minutes.

• Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Vector Spaces”: “Exercise Problem 15” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Vector Spaces”: “Exercise Problem 15”

Also available in:

iBooks

Instructions: Do problem 15 on pages 326 – 327.  The solution can be found on page 407.

• Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Vector Spaces”: “Linear Independence”: “Exercise Problems 1.18, 1.19, and 1.24” Link: St. Michael’s College: Jim Hefferon’s “Linear Algebra” (PDF): “Vector Spaces”: “Linear Independence”: “Exercise Problems 1.18, 1.19, and 1.24”

Also available in:

iBooks

Instructions: Do problems 1.18, 1.19, and 1.24 on pages 106 – 107.  The answers can be found here on pages 49 – 50.

3.4 Change of Basis   - Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Change of Basis” Link: St. Michael's College: Jim Hefferon’s “Linear Algebra” (PDF): “Maps Between Spaces”: “Change of Basis”

`````` Also available in:

– 245.

use displayed on pages iv – vi.  The material linked above is
licensed under the [GNU Free Documentation
attributed to Jim Hefferon and the original version can be
found [here](http://joshua.smcvt.edu/linearalgebra/) (HTML).
``````

Instructions: Please watch the entire lecture, which is about the change of basis matrix.

Watching this lecture should take approximately 20 minutes.

• Assessment: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Vector Spaces”: “Exercise Problems 1.6, 1.7, 1.8, and 1.9” Link: St. Michael’s College: Jim Hefferon’s “Linear Algebra” (PDF): “Vector Spaces”: “Exercise Problems 1.6, 1.7, 1.8, and 1.9”

Also available in:

iBooks

Instructions: Please do problems 1.6, 1.7, 1.8, and 1.9 on page 239.  The solutions to these exercises can be found here on pages 123 – 124.

3.5 Linear Transformations   - Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Computing Linear Maps” Link: St. Michael's College: Jim Hefferon’s “Linear Algebra” (PDF): “Maps Between Spaces”: “Computing Linear Maps”

`````` Also available in:

pages 193 – 203.  This subchapter shows linear transformations as
matrix operations and mappings.  It shows that the two are
interchangeable.

use displayed on pages iv – vi.  The material linked above is
licensed under the [GNU Free Documentation
attributed to Jim Hefferon and the original version can be
found [here](http://joshua.smcvt.edu/linearalgebra/) (HTML).
``````
• Lecture: Harvard University Extension: Dr. Benedict Gross’ “Math E-222 Abstract Algebra”: “Bases and Vectorspaces” Link: Harvard University Extension: Dr. Benedict Gross’ “Math E-222 Abstract Algebra”: “Bases and Vectorspaces” (Flash, QuickTime, or Audio mp3)

Instructions: Please click on the link and then scroll down to Week 4, Lecture 1.  Choose the format most appropriate for your internet connection.  Watch the entire video.

• Assessment: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Vector Spaces”: “Exercise Problems 1.14, 1.15, and 1.17” Link: St. Michael’s College: Jim Hefferon’s “Linear Algebra” (PDF): “Vector Spaces”: “Exercise Problems 1.14, 1.15, and 1.17”

Also available in:

iBooks

Instructions: Do problems 1.14, 1.15, and 1.17 on page 201.  The solutions can be found here on pages 97 – 98.

3.6 Composition of Transformations   - Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Matrix Multiplication” Link: St. Michael's College: Jim Hefferon’s “Linear Algebra” (PDF): “Maps Between Spaces”: “Matrix Multiplication”

`````` Also available in:

pages 213 – 220.  This subchapter demonstrates composition of linear
transformations as matrix multiplication.

use displayed on pages iv – vi.  The material linked above is
licensed under the [GNU Free Documentation
attributed to Jim Hefferon and the original version can be
found [here](http://joshua.smcvt.edu/linearalgebra/) (HTML).
``````
• Lecture: Khan Academy’s “Compositions of Linear Transformation 1” and “Compositions of Linear Transformation 2” Link: Khan Academy’s “Compositions of Linear Transformations 1” and “Compositions of Linear Transformations 2” (YouTube)

Instructions: Please watch both lectures, which cover compositions of linear transformations.

Watching these lectures should take approximately 30 minutes.

• Assessment: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Matrix Multiplication”: “Exercise Problems 2.24, 2.26, and 2.29” Link: St. Michael’s College: Jim Hefferon’s “Linear Algebra” (PDF): “Maps Between Spaces”: “Matrix Multiplication”: “Exercise Problems 2.24, 2.26, and 2.29”

Also available in:

iBooks

Instructions: Do problems 2.24, 2.26, and 2.29 on page 218 – 219.  The solutions can be found here on pages 322 – 323.

3.7 Kernel and Range of Transformations   - Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Rangespace and Nullspace” Link: St. Michael's College: Jim Hefferon’s “Linear Algebra” (PDF): “Maps Between Spaces”: “Rangespace and Nullspace”

`````` Also available in:

Nullspace, pages 181 – 192.

use displayed on pages iv – vi.  The material linked above is
licensed under the [GNU Free Documentation
attributed to Jim Hefferon and the original version can be
found [here](http://joshua.smcvt.edu/linearalgebra/) (HTML).
``````

Instructions: Please watch the lecture, which discusses kernel and preimages for vectors in the range of the transformation.

Watching this lecture should take approximately 15 minutes.

Instructions: Please watch the lecture, which discusses image (or range) of a transformation.

Watching this lecture should take approximately 15 minutes.

• Assessment: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Homomorphisms”: “Rangespace and Nullspace”: “Exercise Problems 2.22, 2.23, 2.24, 2.27, and 2.29” Link: St. Michael’s College: Jim Hefferon’s “Linear Algebra” (PDF): “Homomorphisms”: “Rangespace and Nullspace”: “Exercise Problems 2.22, 2.23, 2.24, 2.27, and 2.29”

Also available in:

iBooks

Instructions: Do problems 2.22, 2.23, 2.24, 2.27, and 2.29 on page 190.  The solutions can be found here on pages 303-304.

3.8 Vector Space Isomorphisms   - Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Isomorphisms” Link: St. Michael's College: Jim Hefferon’s “Linear Algebra” (PDF): “Maps Between Spaces”: “Isomorphisms”

`````` Also available in:

166.

use displayed on pages iv – vi.  The material linked above is
licensed under the [GNU Free Documentation
attributed to Jim Hefferon and the original version can be
found [here](http://joshua.smcvt.edu/linearalgebra/) (HTML).
``````
• Assessment: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Isomorphisms”: “Exercise Problems 1.10, 1.11, 1,13, and 1.14” Link: St. Michael’s College: Jim Hefferon’s “Linear Algebra” (PDF): “Maps Between Spaces”: “Isomorphisms”: “Exercise Problems 1.10, 1.11, 1,13, and 1.14”

Also available in:

iBooks

Instructions: Do problems 1.10, 1.11, 1.13, and 1.14 on page 164.  The answers can be found here on pages 287-290.