Math 202                    Study Guide for Chapter 6                        C.S. Davis

Your are responsible for the subject matter in sections 6.1-6.5 from Elementary Linear Algebra, fifth edition, by Larson and Edwards. The following topics are the most important. Typical exercises are assigned at the end of each topic.  Work more exercises for more practice.

6.1

1.

Define linear transformation (LT) Text

2.

Determine whether a given transformation is linear. P362#7,9,11

3.

Use the terms, domain, codomain and range of the LT. Text

4.

LT's are defined in three basic ways: by rule, by the effect on a basis and by the standard transformation matrix A. Given any one of these, be able to find the image or pre-image of a given vector. P362#1,3,24,25,26

5.

Give examples of rotation, projection, differential and integral LT's. P363#29,31,33,43,44,48

6.2

1.

Define the kernel, range, rank and nullity of a LT.

2.

For a given LT,

a.  find a basis for and/or describe the kernel;

b.  find a basis for and/or describe the range;

c.  find the rank and nullity; P376#1,3,5,13,15,17,21

d.  demonstrate the connection between rank, nullity and the dimension (n) of the domain of a LT. P377#25,27,33,35

e.  use nullity and rank to determine whether a LT is one-to-one, onto and/or an isomorphism. P377#38,39,41

6.3

1.

LT's are defined in three basic ways: by rule, by the effect on a spanning set and by the transformation matrix. Given any one of these, be able to find the other two. Notes

2.

Find the standard transformation matrix for a linear transformation (LT). P387#1,3,9,15,17

3.

Find the standard transformation matrix for a composition of two LT's. P388#25,33

4.

Find the inverse of a LT that is invertible. P388#35,37

5.

Find the transformation matrix relative to bases B and E. P388#41,43,45,47,49

6.

Find T(v) and/or T^-1(w) using 2-5 above. P388#9,15,17

6.4

1.

Find the transformation matrix G for T relative to basis D, given the transformation matrix A for T relative to the standard basis S. P395#1,3

2.

Find the transformation matrix G for T relative to basis D, given the transformation matrix C for T relative to the basis B. P395#7,9

3.

Demonstrate that two transformations matrices are similar.

6.5

1.

Recognize and give linear transformations and transformation matrices for popular geometric transformations in the plane and in space. Include rotation, P403#35-51 odd; contraction, P403#11,15,21; expansion P403#16,22, magnification P403#25,27; reflection P403#9,13,19 and shear P403#17,23