Linear Algebra math help

Math 220 Linear Algebra (Spring 2018)

Homework 1.8 and 1.9

Due Thursday Feb. 15

These will be graded in detail. Be sure to start each of these problems on a new sheet of paper, summarize the problem, and explain what you are doing so that a classmate who is struggling can follow what you are doing by just reading your work (do not refer to any external references, including the text). Some of these problems may be graded on completion only and will be worth only half the number of points indicated.

1. (8 points) Define f : IR ! IR by f(x) = mx + b. (a) (2 points) Write down the two requirements that must be true in order for a transformation

to be linear.

(b) (3 points) Show that f is a linear transformation when b = 0.

(c) (3 points) Find a property of a linear transformation that is violated when b 6= 0.

2. (4 points) Define T : IR3 ! IR2 by

T

0

B @

2

6 4 u1 u2 u3

3

7 5

1

C A =

" u1 + 2 u2u3

#

.

Show that T is not a linear transformation.

3. (4 points) Show that the transformation T : IR2 ! IR3 defined by

T

" x1 x2

#!

=

2

6 4

5x1 x1 � x2 2x2 � x1

3

7 5

is linear.

4. (4 points) Let T : IR3 ! IR3 be a linear transformation. Let {v1, v2, v3} be a set of linearly dependent vectors in IR3. Show that the set {T(v1), T(v2), T(v3)} is also linearly dependent.

5. (10 points) Consider the transformation that first performs a vertical shear that maps e1 to e1 + 2e2 and leaves e2 unchanged and then rotates counterclockwise by 45

�.

(a) Determine the matrix A representing the vertical shear only.

(b) Determine the matrix B that represents rotation by 45� only.

(c) Determine the matrix C that represents the complete transformation (shear followed by a rotation).

(d) Is C = AB, BA, or both? Explain your answer using the geometric interpretation of transformation.