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 φ ∈ End(V) be defined for each f ∈ V by

ϕ(f(t)) = tf′′(t) ,

where f′′ denotes the second derivative of f. What is the matrix of φ 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} − 2x_{2} + 3x_{3} − x_{4} and f_{2}(x) = 2x_{1} − x_{3} + x_{4}.

(a) Find a basis of the the pre-annihilator of W.

(b) 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 × 3 matrices in the field F) for which trace(A) = 0.

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

(b) What is the image of trace : M_{3}(F) ⟶ F?

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

(d) Find the dimension of U without first finding a basis of U.

(e) Find a basis of U.

5. For x, y ∈ F^{n} let

B(x, y) = ∑_{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 λ : V ⟶ V^{∗}.

(a) Prove that this map λ is an isomorphism.

(b) Does the construction of λ 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 α_{V} denotes the natural isomorphism V ⟶ V^{∗∗}, prove that the image under α_{V} of the pre-annihilator of T is the annihilator of T.