Math 220 Assignment

October 29, 2001

Due Wednesday, October 31

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

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

    1. A linearly independent set K of vectors in R^{5} such that every element of the kernel of f is a linear combination of the vectors in K.

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

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

     I_{f}(x)  =  INT[_{0}^{x}  f ]  .  

    1. Explain briefly why I_{f} is abstractly linear.

    2. What is the kernel of I_{f} ?

    3. In what set does the function I_{f} takes its values? (The domain of I_{f} is understood here to be \cal{P}_{d}. )

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