Math 220 Assignment

December 3, 2001

Relevant Sections of the Course Textbook

Chapter 1
§§ 1-7

Chapter 2
§§ 1-8

Chapter 3
§§ 1-5

Chapter 4
§§ 1-3, 5

Chapter 5
§§ 1, 2

Chapter 6
§§ 1, 2, 4-7

(Additionally, there has been passing mention of § 4 in Chapter 4 and of topics in Chapter 7. There will be no testing on these supplementary topics.)

Due Wednesday, December 5

The geometric significance of an n \times n matrix that is similar to a diagonal matrix is that the corresponding linear transformation carries each line in some set of n lines, in ``general position'' passing through the origin, to itself.

  1. What is a more precise description of the phrase general position in the preceding statement?

  2. What is the set of n lines when n = 2 and the matrix is

     
    (
    5 
    12
    -12
    5
    		)
    		  ?  

  3. Find an example with n = 2 where the said set of 2 lines in general position is a pair of non-parallel lines through the origin that, instead of being perpendicular, form the angle pi/4 (i.e., 45 degrees) at the origin.

  4. Show that the matrix

     
    (
    2 
    1
    0
    2
    		)
    		 
    is not similar to a diagonal matrix.

  5. What geometric property might be said to characterize the n \times n matrices that are similar to upper triangular matrices?

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