Let W, Z be complementary subspaces of a vector space V, as in Exercise 2.2.24. Let V/W denote the quotient vector space, as defined in Exercise 2.2.29. Show that the map L:Z → V/W that maps L[z] = [z]_W defines an invertible linear map, and hence Z ≃ V/W are isomorphic vector spaces.

Exercise 2.2.24

Let W ⊂ V be a subspace. A subspace Z ⊂ V is called a complementary subspace to W if 

(i) W ∩ Z = {0}, and 

(ii) W + Z = V , i.e., every v ∈ V can be written as v = w + z for w ∈ W and z ∈ Z.

(a) Show that the x- and y-axes are complementary subspaces of R².
(b) Show that the lines x = y and x = 3y are complementary subspaces of R².

(c) Show that the line (a, 2a, 3a)^T and the plane x + 2y + 3z = 0 are complementary subspaces of R³. 

(d) Prove that if v = w + z, then w ∈ W and z ∈ Z are uniquely determined.

Exercise 2.2.29

Let V be a vector space and W ⊂ V a subspace. We say that two vectors u, v ∈ V are equivalent modulo W if u − v ∈ W. 

(a) Show that this defines an equivalence relation, written u ∼ _W v on V , i.e.,

(i) v ∼ _W v for every v;
(ii) if u ∼ _W v, then v ∼ _W u; and 

(iii) if, in addition, v ∼ _W z, then u ∼ _W z. 

(b) The equivalence class of a vector u ∈ V is defined as the set of all equivalent vectors, written [u]_W = { v ∈ V | v ∼ _W u }. Show that [0]_W = W. 

(c) Let V = R² and W = {( x, y )^T | x = 2y}. Sketch a picture of several equivalence classes as subsets of R². 

(d) Show that each equivalence class [u]_W for u ∈ V is an affine subspace of V.

(e) Prove that the set of equivalence classes, called the quotient space and denoted by V/W = {[u] | u ∈ V}, forms a vector space under the operations of addition, [u]_W + [v]_W = [u + v]_W, and scalar multiplication, c[u]_W = [cu]_W. What is the zero element? Thus, you first need to prove that these operations are well defined, and then demonstrate the vector space axioms.