Advanced Linear Algebra (Math 424/524)
Assignment No. 5

Due December 11, 2002

  1. Let F be a field. For P a polynomial in F[t] let V_{P} denote the quotient F[t]/P F[t]. Given P and Q in F[t] one defines a linear map

     phi : V_{PQ}  ----->  V_{P} \times V_{Q} 

    by

     phi(hmod PQ)  =  ( hmod P,  hmod Q)  . 
    What is the dimension of the kernel of phi ?

  2. For each of the following rational matrices find the minimal and characteristic polynomials and find a direct sum of companion matrices that is similar to the given matrix:

    (a)
    (
    1 
    1
    0
    1
    		)
    		(b)
    (
    0 
    1
    -1
    0
    		)
    		(c)
    (
    0 
    1
    0
    0
    0
    -1
    1
    0
    0
    		)

    (d)
    (
    0
    1
    0
    0
    0
    1
    1
    0
    0
    		)
    		(e)
    (
    1 
    2
    0
    -1
    3
    0
    0
    -2
    1
    -1
    -1
    3
    1
    0
    0
    2
    -1
    3
    -1
    2
    1
    -1
    -3
    -2
    0
    		)

  3. Let M be the matrix of polynomials in F[t]

     M  =  
    (
    t^{3}+t^{2}-2t 
    t^{3}-2t+1
    t^{3}-1
    t^{3}-t^{2}
    		)
    		  . 

    1. Find a diagonal matrix of successively divisible polynomials that can be obtained from M by (restricted) row and column operations.

    2. What is the dimension of the quotient space

        F[t]^{2}/M F[t]^{2}  ? 

  4. Let M be the matrix of polynomials from the previous problem.

    1. Is there a 2 \times 2 matrix A for which t. 1 - A is (restricted) row and column equivalent to M ?

    2. Find an N \times N matrix A for some N in the field F for which the endomorphism “multiplication by t” of the quotient space in part (b) of the previous problem is isomorphic to the endomorphism of F^{N} given by A.

  5. For given lambda in the field F let J = J(lambda) be the r \times r matrix

     
    (
    lambda 
    1
    0
    0
    ...
    0
    0
    lambda
    1
    0
    ...
    0
    0
    0
    lambda
    1
    :
    :
    :
    0
     
    lambda
    1
    0
    0
    0
    ...
    0
    lambda
    		)
    		  . 
    Find a basis for the vector space F[t]/(t - lambda)^{r} F[t] in which the matrix J represents the endomorphism “multiplication by t”.

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