I) Vector Spaces and Subspaces;

Module 2/Week 2

·  Module Overview

A. We will assume that the concepts and techniques from a first linear algebra course have been reviewed. The most important concepts and techniques, in terms of their use in the upcoming material, are:

i)  Vector spaces and subspaces;

ii)  Linearly independent sets and basis;

iii)  Basis and dimension of vector spaces;

iv)  Null Space, Column Space, Row Space and Rank of a matrix;

v)  Solution sets of Linear systems of equations;

vi)  Matrix Algebra.

B.  We will answer questions about the first weeks’ assignments, which are related to the review of the introductory course, as well as to the use of Matlab.

C.  We will study section 4.1, which will cover the definition and properties of linear transformations.

·  Content for week 2: Linear Transformations.

A.  We will see the definition of a linear transformation.

B.  We will next study examples. The first examples will be of linear transformations (operators) from; then we will study examples in which the domain and co-domain can be any combination of Euclidean () and general vector spaces (). We will show that each matrix defines a linear transformation from.

C.  We will also study the kernel and range of linear transformations, as well as injective (one-to-one) and surjective (onto) linear transformations.

·  Goals and Objectives:

1) Students will be able to recognize a linear transformation.

2) Students will be able to find the kernel and range of a linear transformation.

1)  Recognize and prove that a given mapping is a linear transformation or is not.

2)  Find the kernel and range of a linear transformation.

3)  Determine if a linear transformation is one-to-one, onto, or neither.

4)  Show that each matrix defines a linear transformation from.

·  Assignments and Activities:

A. Read and Watch

(i) Read Section 4.1, pages 166-174, of Leon.

(ii) Lesson with tablet

B. Assignments

Assignment #1: Do exercises Leon: pages 174-175

# 1,3,4,5,6,7,9,10,11,13,14,17,19,21,23,

Assignment #2: Do Matlab-Atlast exercise:

Page 88, #2(b) #3 (a) and (b).

C. Forum: Feel free to share your answers and work together. Those who want to post the first answer to a problem, or an alternative answer to a problem already posted, will have it taken into account for participation credit.

Internet Search Assignment: Do a Google search for “Linear Transformation” and share with the class at least one definition (or explanation), and its source, that complements the material in the book.

D. Recommended Supplements for this Section. This material is not mandatory. There are some differences in terminology and notation in the different texts. As this is very common, even amongst high school algebra, geometry, precalculus and calculus textbooks, it is very good practice for real life issues in teaching mathematics at any level, or reading articles on mathematics and mathematics education.

:

(i) Read Chapter 3, sections 2 and 3 from the online text : http://joshua.smcvt.edu/linalg.html/toc.html . Do as many problems as possible. The book has solutions to almost all problems.

(ii) Read Chapter LT, all sections, and do as many problems as possible from the online text: http://linear.ups.edu/online.html

(iii) Watch MIT video: Lecture #30: Linear Transformations and Their Matrices http://web.mit.edu/18.06/www/Video/video-fall-99.html

(iv) Read notes about Linear Transformations in Euclidean Spaces in the section on Euclidean n-spaces in http://tutorial.math.lamar.edu/Classes/LinAlg/LinearTransformations.aspx