Higher Algebra 2 — Spring 2006

Gerald Hoehn

Problem sheet 1 January 20, 2006

Problem 1: (a) Prove that any two bases in a vector space have the same cardinality. (Hint: Generalize the case of a vector space with finite a base (Theorem 2). Use the fact that a finite number of elements in one basis is contained in the space generated by a finite number of elements in another basis.)

(b) Prove that if the dimension of a vector space V is infinite, the cardinality of a basis in the dual space V^{∗} is larger the the cardinality of a basis of V.

Problem 2: Let V be a vector space over a field K, and let U, W be subspaces. Show that

dim U + dim W = dim(U + W) + dim(U ∩ W).

Here, U + W denotes the subspace of linear combinations of elements of U ∪ W.

Problem 3^{∗∗∗}: For a vector space V over a field K, denote by i_{V} : V ⟶ V^{∗∗} the canonical injection given by evaluation and by i_{V}^{∗} : V^{∗∗∗} ⟶ V^{∗} its adjoint.

(a) Prove that i_{V}^{∗} ∘ i_{V^{∗}} : V^{∗} ⟶ V^{∗} equals the identity map of V^{∗}.

(b) For a base B of V denote by B^{∗} be the dual base. Let f_{B} : V ⟶ V^{∗} be the injection defined by using the base B and f_{B^{∗}} : imf_{B} ⟶ (imf_{B})^{∗}. Prove that

res_{imf_{B}}^{V^{∗}} ∘ i_{V} = f_{B^{∗}} ∘ f_{B},

where res_{imf_{B}}^{V^{∗}} is the natural restriction of a linear functional on V^{∗} to imf_{B}.

Problem 4: Determine the number of k-dimensional subspaces of an n-dimensional vector space over a field with q elements.