Let V be a vector space over K and let W1 and W2 be subspaces of V. Prove that \[ \frac{W1 + W2}{W2} \cong \frac{W1}{W1 \cap W2} \]

Steps to Solve

1. Define the Quotient Spaces

   Let ( V ) be a vector space and ( W_1, W_2 ) be subspaces of ( V ). The sum of the subspaces is defined as: $$ W_1 + W_2 = { w_1 + w_2 \mid w_1 \in W_1, w_2 \in W_2 } $$

   The quotient space ( \frac{W_1 + W_2}{W_2} ) consists of the cosets of ( W_2 ) in ( W_1 + W_2 ).

2. Understanding the Isomorphism

   We want to demonstrate that: $$ \frac{W_1 + W_2}{W_2} \cong \frac{W_1}{W_1 \cap W_2} $$

   We will use the projection map from ( W_1 + W_2 ) onto ( W_1 ) modulo ( W_1 \cap W_2 ).

3. Defining the Isomorphism

   Define a map ( \phi: W_1 + W_2 \to W_1 ) by: $$ \phi(w_1 + w_2) = w_1 $$ where ( w_1 \in W_1 ) and ( w_2 \in W_2 ).

   This map is well-defined and linear since it respects vector addition and scalar multiplication.

4. Kernel of the Map

   The kernel of ( \phi ) is: $$ \ker(\phi) = { w_1 + w_2 \in W_1 + W_2 \mid w_1 \in W_1, w_2 \in W_2 \text{ such that } \phi(w_1 + w_2) = 0 } $$ This corresponds to: $$ \ker(\phi) = W_1 \cap W_2 $$

5. Applying the First Isomorphism Theorem

   By the First Isomorphism Theorem for vector spaces, we have: $$ \frac{W_1 + W_2}{\ker(\phi)} \cong \text{Image}(\phi) $$ Hence: $$ \frac{W_1 + W_2}{W_1 \cap W_2} \cong W_1 $$

   Now, consider the map ( \psi: W_1 \to \frac{W_1}{W_1 \cap W_2} ): $$ \psi(w_1) = [w_1] $$

   So we conclude: $$ \frac{W_1 + W_2}{W_2} \cong \frac{W_1}{W_1 \cap W_2} $$