Math 220 Assignment

October 31, 2001

Due Friday, November 2

  1. Let g be the linear function from R^{4} to R^{5} that is defined by g(x) = M x where M is the 5 \times 4 matrix

     
    (
    -1 
    1
    5
    1
    2
    -1
    2
    1
    1
    0
    -2
    2
    -2
    2
    1
    2
    -4
    3
    8
    1
    		)
    		  .  
    Find the following:

    1. A basis for the kernel of g.

    2. A non-redundant list of linear equations that characterize the image of g as a subset of R^{5}.

    3. A basis for the image of g.

  2. Let \cal{P}_{2} denote the vector space of polynomials of degree 2 or less. If f is an element of \cal{P}_{2}, let T_{f} be the polynomial given by the formula

     T_{f}(x)  =  {d}/{dx} x f(x)  .  

    1. Show that the function T that is defined by

       T(f)  =  T_{f} 
      is an abstractly linear map from \cal{P}_{2} to \cal{P}_{2}.

    2. What is the dimension of \cal{P}_{2} ?

    3. Find a basis of the kernel of T.

    4. Find a basis of the image of T.

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