Matrices and Linear Transformations

MA 2071

Thursday  April 17, 2003

 

 

You need to know that the matrix A of a linear transformation L  (  so L(x) = Ax )

is constructed by figuring out what L does to the standard basis for its domain and then using those vectors as the columns of A.

 

 

  1. a)  If  L rotates vectors  45 degrees clockwise, what is the matrix of L ?

      b)  suppose x =  the vector (1,4)    Neatly sketch this and then multiply it by A

           from part a) and sketch that as well.  Essentially see if things work as they

           should.

c)      What should  A4  be?

d)      What should  A8  be?

e)      What does  A-1 do?       A-2 ?

 

 

2        a)  If  L is a counterclockwise rotation, use basic geometry to construct

             its matrix, A.

 

     b)  What does the matrix A2 do?

compute A2  and compare with the matrix derived in class today

 

     c)   What is A12 ?   A240 ?

 

3.  Suppose  B is the matrix for a 60 degree clockwise rotation and A the matrix

for a 30 degree clockwise rotation .  Argue why A can be thought of as the

square root of B.     (yes, it’s possible for matrices to have square roots!)

 

 

4.   Set up a general matrix , call it A, which would rotate vectors counterclockwise through an angle  q.  Set up another matrix B which similarly would rotate vectors counterclockwise through an angle  a.

 

     a)  what would the matrix  AB do?

     b)  why, in this case, should  AB = BA  ?

    c)  what trig identities can you derive by examining AB ?

 

5.  if determinants give you areas of parallelograms determined by the vectors making up their columns, argue why the determinant of any matrix of rotation should be +/- 1.  Why should the columns be orthonormal?  Unit vectors?