MA20008 Algebra 1, 2004, Sheet 5

Geoff Smith, http://www.bath.ac.uk/∼masgcs

1. Let V be a vector space of dimension n.

(a) Suppose that we have subspaces

0 = V_{0} < V_{1} < V_{2} ⋯ < V_{n} = V

with dim V_{i} = i for every i = 0, 1, …, n. For each i > 0 choose v_{i} ∈ V_{i} but v_{i} ∉ V_{i−1}. Show that

v_{1}, v_{2}, …, v_{n}

is a basis of V.

(b) Suppose that

v_{1}, v_{2}, …, v_{n}

is a basis of V. Show that there are subspaces

0 = V_{0} < V_{1} < V_{2} ⋯ < V_{n} = V

with dim V_{i} = i for every i = 0, 1, …, n such that v_{i} ∈ V_{i} but v_{i} ∉ V_{i−1} for every i > 0.

2. Let V be a vector space of dimension n. Suppose that α : V ⟶ V is a linear map.

(a) Show that V ≥ Im α ≥ Im α^{2} ≥ … ≥ 0. Observation: Im α^{r} denotes the map defined by composing r copies of α.

(b) Show that 0 ≤ Ker α ≤ Ker α^{2} ≤ ⋯ ≤ V.

(c) Suppose that r is a natural number and Im α^{r} = Im α^{r+1}. Show that Im α^{t} = Im α^{r} for all natural numbers t ≥ r.

(d) Let V be a vector space of dimension n. Suppose that α : V ⟶ V is a linear map such that α^{2} = 0 (the zero map). Show that the nullity ν_{α} satisfies 2ν_{α} ≥ n.

(e) Suppose that there is a natural number m such that α^{m} is the zero map 0. Prove that α^{n} is 0.

3. Suppose that V is a vector space of dimension n and that α : V ⟶ V is a linear map. Suppose that α^{3} is 0, the zero map. Show that ν_{α} ≥ n/3 where ν_{α} denotes the nullity of α. Hint: let V_{0} = V, V_{1} = Im α and V_{2} = Im α^{2}. Let α_{1} : V ⟶ V_{1} and α_{2} : V_{1} ⟶ V_{2} be the maps defined by α_{i}(v) = α(v) for all v ∈ V_{i−1}. Deploy the rank nullity theorem.

4. Let V be a finite dimensional vector space, with W, X, Y and Z all subspaces of V.

(a) Show that

dim(X ∩ Y) + dim((X + Y) ∩ Z)
= dim(Y ∩ Z) + dim((Y + Z) ∩ X)
= dim(Z ∩ X) + dim((Z + X) ∩ Y).

(b) Show that

dim(W ∩ X) + dim(Y ∩ Z) + dim((W + X) ∩ (Y + Z))

= dim(W ∩ Y) + dim(X ∩ Z) + dim((W + Y) ∩ (X + Z))
= dim(W ∩ Z) + dim(X ∩ Y) + dim((W + Z) ∩ (X + Y))

Hint: this question is not hard. You need an idea. Please do not start picking bases; that way madness lies.