Advanced Linear Algebra
Math 424/524

Assignment No. 3

Due November 13, 2002

  1. Find a 2 \times 2 real diagonal matrix D for which there exists a matrix U that is orthogonal relative to the standard inner product (the “dot” product) on R^{2} and satisfying

      U^{t} 
    (
    1 
    2
    2
    -1
    		)
    		 U   =   D  .  

  2. Find an invertible 2 \times 2 matrix U of rational numbers and a rational diagonal matrix D such that

      U^{t} 
    (
    1 
    2
    2
    -1
    		)
    		 U   =   D  .  

  3. When 2 <> 0 in the field F, find an invertible 2 \times 2 matrix U in F such that

      U^{t} 
    (
    0 
    1
    1
    0
    		)
    		 U  =  
    (
    1/9 
    0
    0
    -1/9
    		)
    		  .  

  4. Find a 3 \times 3 rational matrix that is orthogonal for the standard inner product on R^{3} with the property that none of its entries has absolute value 1.

  5. Find an invertible matrix U such that

     U^{t} 
    (
    0 
    0
    -1
    0
    0
    0
    0
    -1
    1
    0
    0
    0
    0
    1
    0
    0
    		)
    		  U  =   
    (
    0 
    1
    0
    0
    -1
    0
    0
    0
    0
    0
    0
    1
    0
    0
    -1
    0
    		)
    		  .  

  6. What conditions in the definition of inner product are not satisfied by the bilinear form Theta on the space M_{n}(R) of n \times n real matrices defined by

     Theta(M, N)  =  trace
    (M N)
    		  .  

  7. For a field F in which 2 = 0 give an example of a bilinear form on F^{3} that is skew-symmetric but not alternating.

  8. Let b be the bilinear form on F^{3} given by

      b(x, y)  =  x^{t} 
    (
    0 
    1
    0
    0
    0
    1
    1
    0
    0
    		)
    		 y  .  

    1. Explain very briefly why b is dualizing.

    2. Find the left orthogonal complement of U, i.e.,

       
      { v \in F^{3}  |    b(v, u)  =  0  for each   u \in U}
      		 
      in each of the three cases when U is a coordinate axis.

    3. When 2 <> 0 in F, find a symmetric bilinear form s and an alternating bilinear form a on F^{3} such that b = s + a.

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