HOMEWORK 10 — DUE DEC 5TH

MATH 25

There are two sections: A and B. Please return each part to the proper CA.

A. Problems graded by Benjamin

A.1. Find the rank of the matrix A = (a_{i,j})_{1≤i,j≤n} in the following cases:

(1) a_{i,j} = i + j − 1.

(2) a_{i,j} = cos(α(i + j)) where α ∈ R.

(3) a_{i,j} = (α + i + j)^{2} where α ∈ R.

A.2. Let E be a vector space over a field K, and let F be a subspace (not equal to E). Prove that Vect(E \ F) = E (i.e. that every vector in E is a linear combination of vectors of E \ F). Make sure your proof works for every field K.

A.3. Let K be an infinite field, let V be a K-vector space of dimension n and let W_{1}, ⋯, W_{k} be subspaces of V of dimension ≤ n − 1. Prove that V ≠ ∪_{i=1}^{k}W_{i}.

Show that this is not always true if K is finite.

A.4. Let V be a K-vector space of dimension n and let W_{1} and W_{2} be two subspaces of the same dimension r ≤ n − 1. Prove that there exists a subspace U of V such that V = W_{1} ⊕ U and V = W_{2} ⊕ U.

A.5. Let V be a finite dimensional K-vector space and let u, v, w ∈ L(V, V) be three endomorphisms of V. Prove the following inequalities:

(1) rk(u + v) ≤ rk(u) + rk(v).

(2) rk(u + v) ≥ | rk(u) − rk(v)|.

(3) inf(rk(u), rk(v)) ≥ rk(uv) ≥ rk(u) + rk(v) − dim(V)

(4) rk(uv) + rk(vw) ≤ rk(v) + rk(uvw). Use the last inequality to prove that if k ≥ 1, and u^{k} denotes u × ⋯ × u (k times) then:

rk(u^{k}) ≤ \frac{1}{2}(rk(u^{k+1}) + rk(u^{k−1})).

2 MATH 25

B. Problems graded by Inna

B.1. Let e = {e_{1}, ⋯, e_{n}} and f = {f_{1}, ⋯, f_{n}} be two bases of a vector space V. Check that if A is the matrix of e in f and if B is the matrix of f in e then AB = BA = Id.

B.2. Let V be the space of polynomials P ∈ R[X] whose degree is at most n − 1.

(1) check that {1, X, X^{2}, ⋯, X^{n−1}} is a basis of V and that the map P (X) ↦ P (X + 1) is a linear map from V to itself.

(2) Write the matrix M of P (X) ↦ P (X + 1) in the above basis.

(3) find M^{−1}.

B.3. Let V be the vector space of all functions from R to R. Prove that the following families are free:

(1) {e_{1} ⋯ e_{n}} where the α_{i}’s are distinct real numbers and e_{i}(x) = e^{α_{i}x}.

(2) {f_{1} ⋯ f_{n}} where the α_{i}’s are distinct real numbers and f_{i}(x) = |x − α_{i}|.

B.4. Let U, V, W be three (finite dimensional) K-vector spaces, and let w ∈ L(U, W).

(1) If u ∈ L(U, V), prove that there exists v ∈ L(V, W) such that w = vu if and only if ker u ⊂ ker w.

(2) If v ∈ L(V, W), prove that there exists u ∈ L(U, V) such that w = vu if and only if im w ⊂ im v.

B.5. Let M be a n × n matrix with entries in a field K.

(1) prove that M is invertible if and only if it has rank n.

(2) prove that if we write M = (C_{1}, ⋯, C_{n}) then M is invertible if and only if the dimension of the subspace of K^{n} generated by C_{1}, ⋯, C_{k} is k for every 1 ≤ k ≤ n.

(3) assume now that K = Z/pZ, the field with p elements. How many n × n matrices with entries in Z/pZ are there?

(4) how many of them are invertible?

(5) (extra credit) let f(n, r) be the number of n × n matrices with entries in Z/pZ which have rank r. Give a formula for f(n, r) and check that f(n, 0)+f(n, 1)+⋯+f(n, n) agrees with the result of (3).