Exercise 1:

Let V and W be vector spaces over a field F. Let α ∈ Hom(V, W) and β ∈ Hom(W, V) satisfy the condition that αβα = α.

If ω ∈ im(α), show that

α^-1(ω) = {β(ω) + v - βα(v)|v∈V}

Exercise 2:

Let

α : R3 → R3 be the linear transformation given by

α :

Find ker(α) and im(α).

Exercise 3:

Let V be a finite-dimensional vector space over a field F and let α, β ∈ Hom(V, V) be linear transformations satisfying im(α) + im(β) = V = ker(α) + ker(β). Show that im(α) ∩ im(β) = {0V} = ker(α) ∩ ker(β).

Exercise 4:

Let V be a finitely-generated vector space over a field F and let α ∈ End(V).

Show that α is not monic if and only if there exists an endomorphism β ≠ σ_o.

V satisfying αβ = σ₀.

Exercise 5:

Let V be a vector space over a field F and let α ∈ End(V). Show that ker(α) = ker(α²) if and only if ker(α) and im(α) are disjoint.

Exercise 6:

Let α and β be endomorphisms of a vector space V over a field F satisfying αβ = βα. Is ker(α) invariant under β?

Exercise 7:

Let V be a vector space over a field F and let α,β ∈ End(V). Show that α and β are projections satisfying ker(α) = ker(β) if and only if αβ = α and βα = β.

Exercise 8:

Let B = {1 + i, 2 +i}, which is a basis for C as a vector space over R. Let α be the endomorphism of this space defined by α: z-> z^-. Find Φ_BB(α).

Exercise 9:

Find a nonzero matrix A in M_2x2(R) satisfying v ?. Av = 0 for all v ∈ R².