189-251B: Algebra 2 Assignment 5

Due: Wednesday, February 12.

1. Suppose that T : V ⟶ V is a linear transformation of vector spaces over R whose minimal polynomial has no multiple roots. Show that V can be expressed as a direct sum

V = V_{1} ⊕ V_{2} ⊕ ⋯ ⊕ V_{t}

of T -stable subspaces of dimensions at most 2. Show that, relative to a suitable basis, T can be represented by an n × n matrix with at most 2n non-zero entries, where n := dim(V).

2. Let g(x) be a polynomial in F [x] of degree d and let V be the ring F [x]/(g(x)).

(a) Show that V is a d-dimensional vector space over F.

(b) Let T be the function on V defined by the rule T (v) = [x]v. (Where [x] denotes the equivalence class of x in the quotient ring.) What is the minimal polynomial of T? When is T diagonalisable?

3. Let A be an upper-triangular matrix with entries in a field F. Show that A is diagonalisable if all its diagonal entries are distinct.

4. Let A be an upper-triangular matrix with entries in a field F. Suppose that all the diagonal entries of A are equal. Show that A is diagonalisable if and only if it is diagonal.

5. Let A be an invertible matrix with entries in F = Z_{p}. Show that A is diagonalisable if and only if its order (i.e., the least t with A^{t} = 1 in the group GL_{n}(Z_{p}) of n × n matrices with entries in Z_{p}) divides p − 1.

6. Let V be the (infinite-dimensional) vector space of infinitely differentiable real-valued functions on the real line, and let W be the subset of V consisting of functions f satisfying the differential equation

a_{n}f^{(n)} + a_{n−1}f^{(n−1)} + ⋯ + a_{1}f′ + a_{0}f = 0,

where the scalars a_{j} are real. (Here, f^{(j)} denotes the j-th derivative of f.)

(a) Show that W is a vector subspace of V. (This is why this equation is called a linear differential equation!)

(b) Let T be the function from V to V defined by T (f) = f′. Show that T is a linear transformation that maps W to itself.

(c) From now on, use T to denote the restriction of T to the space W. Show that the eigenvalues of T are precisely the roots of the polynomial g(x) = a_{n}x^{n} + a_{n−1}x^{n−1} + ⋯ + a_{1}x + a_{0}.

(d) Show that the eigenspace for T attached to an eigenvalue λ is always one-dimensional.

(e) Suppose that (x − λ)^{r} divides g(x) exactly (i.e., no higher power of (x − λ) divides it). Show that the generalised eigenspace for T attached to the eigenvalue λ is of dimension r and write down a basis f_{1}, …, f_{r} for this generalised eigenspace such that f_{j} belongs to the kernel of (T − λ)^{j} but not of (T − λ)^{j−1}. (You may use without proof all that you know about linear differential equations of first order.)