PROBLEM SET #2

DUE OCTOBER 6

1. Basic Properties of the Determinant. In this problem, let A be an n × n matrix over 𝔽.

(1) Using the definition, show that if A has two rows which are scalar multiples of each other (so for some i and k, a_{i,j} = λa_{k,j} for all j), then |A| = 0.

(2) Show that |A| = |A^{t}|.

(3) Again using the definition, show that if we swap two rows or columns in a matrix, then we swap the sign of the determinant (you may use without proof the following facts: sign(σ ∘ τ) = sign(σ)sign(τ), and if f_{i,j} is the bijection from {1, …, n} to itself that swaps i and j and fixes everything else, then sign(f_{i,j}) = −1.)

(4) Show A · adj(A) = det(A)I_{n} = adj(A) · A. 5* This problem is optional! Justify the results about the sign representation

used in 3. The basic idea is that anything can be written as a product of transpositions f_{i,j} (and as a fun pop-culture reference, this group and these ideas occurred in a recent episode of Futurama).

2. Let U, V, W be vector spaces over 𝔽. This problem and the following will explore the “categorical” properties of finite direct sums=finite direct products.

(1) Let π_{U}: U⊕V → U and π_{V} : U⊕V → V be the projection maps π_{U}(u⃗+v⃗) = u⃗ (resp for V). Show that these are surjective linear transformations with ker(π_{V}) = U and ker(π_{U}) = V.

(2) Show that we have a natural isomorphism

L(W, U ⊕ V) → L(W, U) ⊕ L(W, V)

given by T ↦ π_{U} ∘ T ⊕ π_{V} ∘ T.

(3) Now let i_{U} (resp i_{V}) denote the inclusion U → U ⊕ V. Show that these are injective linear transformations and π_{V} ∘ i_{U} = 0, π_{U} ∘ i_{V} = 0.

(4) Show that we have a natural isomorphism

L(U ⊕ V, W) → L(U, W) ⊕ L(V, W)

given by T ↦ T ∘ i_{U} ⊕ T ∘ i_{V}.

3. This problem will play with the notion of “equal” versus “isomorphic”. It also stresses some of the pathologies with infinite dimensional spaces. Let ℝ[x] be the vector space of all polynomials in x over ℝ.

(1) Show that ℝ[x] is infinite dimensional.

(2) Show that the subspace of all polyomials which have a root at 0 is isomorphic to ℝ[x]. Conclude that if we just know V ⊕ W ≅ V ⊕ U, then we can’t conclude that W ≅ U.

(3) Show that the derivative is a surjective map ℝ[x] → ℝ[x]. Compute the kernel. Conclude that a surjective map between infinite dimensional vector spaces need not be an isomorphism.

(4) Show that the integral ∫_{0}^{x} provides a one-sided inverse to the derivative. Determine the image. Conclude that an injective map between infinite dimensional vector spaces need not be an isomorphism.

(5) There is a standard basis for ℝ[x]: the basis of all monic monomials in x. If we mimic the arguments given in class, then we can associate an ∞ × ∞- matrix to any linear transformation (indexed by ℕ × ℕ, so really it has a well-defined “upper left corner”). Find the matrix for the derivative and integral.

(6) Building on the previous part, does any ∞ × ∞-matrix (again, indexed by ℕ × ℕ) occur as the matrix associated to a linear operator? What is the image of the map that sends a linear operator to its matrix?