MAT 424/524
Midterm WarmUp

Informal Exercises

October 16, 2002

  1. Let V be the vector space of all polynomials of degree at most 3 with coefficients in the field F. Let phi \in End(V) be defined for each f \in V by

     phi(f(t))  =  t f''(t)  , 
    where f'' denotes the second derivative of f. What is the matrix of phi with respect to the basis { 1, t, t^{2}, t^{3} } of V ?

  2. Let R^{4} denote 4-dimensional column space over the field R of real numbers. Let S be the subset of R^{4} consisting of the two vectors v_{1} = (2, -1, -1, 1) and v_{2} = (1, -2, 4, 2), and let W be the subspace of the dual space of R^{4} spanned by the two linear forms f_{1}(x) = x_{1} - 2 x_{2} + 3 x_{3} - x_{4} and f_{2}(x) = 2 x_{1} - x_{3} + x_{4}.

    1. Find a basis of the the pre-annihilator of W.

    2. Find a basis of the annihilator of S.

  3. Let F be a field, and let P(t) be a member of the ring F[t] of polynomials with coefficients in F. What is the dimension of the quotient space

      F[t]/P(t)F[t]  ?  

  4. Let U be the set of matrices A in the vector space M_{3}(F) (of all 3 \times 3 matrices in the field F) for which trace(A) = 0.

    1. Show that trace: M_{3}(F) -----> F is a linear map.

    2. What is the image of trace: M_{3}(F) -----> F ?

    3. Explain why U is a linear subspace of M_{3}(F).

    4. Find the dimension of U\@ without first finding a basis of U.

    5. Find a basis of U.

  5. For x, y \in F^{n} let

     B(x, y)  =  SUM_{i = 1}^{n}[ x_{i} y_{i} ]  .  
    Observe that for each y the map x -> B(x, y) is a linear form on V, and, therefore, the map y -> B( , y) is a linear map lambda : V -----> V^{*}.

    1. Prove that this map lambda is an isomorphism.

    2. Does the construction of lambda involve choice?

  6. Let V be a finite-dimensional vector space over a field F, and let T be a subset of the dual space V^{*}. Recall that T has a pre-annihilator that is a subspace of V and also an annihilator that is a subspace of the second dual V^{**}. If alpha_{V} denotes the natural isomorphism V -----> V^{**}, prove that the image under alpha_{V} of the pre-annihilator of T is the annihilator of T.

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