Math 220 Assignment

October 1, 2001

Due Wednesday, October 3

  1. Let f be the linear map given by f(x) = M x where M is the matrix

     
    (
    1 
    5
    -2
    -2
    4
    -3
    -1
    -3
    1
    		)
    		  .  

    1. Find the fibre of f over the origin.

    2. Find the fibre of f over the point ( 1, -5, 3).

    3. Find the fibre of f over the point (-1, 2, 1).

    4. Find the set of all points y of R^{3} for which the fibre of f over y is non-empty.

  2. Let g be the linear map given by g(y) = N y where N is the matrix

     
    (
    1 
    2
    0
    2
    -2
    -1
    3
    2
    -2
    2
    6
    -1
    1
    0
    -2
    0
    		)
    		  .  

    1. Find the fibre g^{-1}(0).

    2. Find the fibre g^{-1}(1, -2, -2, 1).

    3. Find the fibre g^{-1}(-1, -7, 2, 1).

    4. Find equations that characterize the set of all x in R^{4} for which g^{-1}(x) is non-empty.

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