1. **Problem Statement:** (a) Define what it means for a vector space $V$ to be the direct sum of two subspaces $W_1$ and $W_2$, denoted as $V = W_1 \oplus W_2$. (b) Provide an example in $\mathbb{R}^3$ where $V = W_1 \oplus W_2$, clearly defining $W_1$ and $W_2$. 2. **Definition of Direct Sum:** The vector space $V$ is the direct sum of subspaces $W_1$ and $W_2$, written as $V = W_1 \oplus W_2$, if every vector $v \in V$ can be uniquely written as $$v = w_1 + w_2$$ where $w_1 \in W_1$ and $w_2 \in W_2$, and the intersection of $W_1$ and $W_2$ contains only the zero vector: $$W_1 \cap W_2 = \{0\}$$ This means the sum is direct because the decomposition of any vector into components from $W_1$ and $W_2$ is unique. 3. **Example in $\mathbb{R}^3$:** Let $V = \mathbb{R}^3$. Define subspaces: $$W_1 = \{(x,0,0) : x \in \mathbb{R}\}$$ $$W_2 = \{(0,y,z) : y,z \in \mathbb{R}\}$$ 4. **Verification:** - Every vector $v = (x,y,z) \in \mathbb{R}^3$ can be written as $$v = (x,0,0) + (0,y,z)$$ where $(x,0,0) \in W_1$ and $(0,y,z) \in W_2$. - The intersection $W_1 \cap W_2 = \{(0,0,0)\}$ because the only vector common to both is the zero vector. 5. **Conclusion:** Thus, $\mathbb{R}^3 = W_1 \oplus W_2$ with the given $W_1$ and $W_2$. **Final answer:** (a) $V = W_1 \oplus W_2$ means every $v \in V$ can be uniquely expressed as $v = w_1 + w_2$ with $w_1 \in W_1$, $w_2 \in W_2$, and $W_1 \cap W_2 = \{0\}$. (b) Example: $V = \mathbb{R}^3$, $W_1 = \{(x,0,0)\}$, $W_2 = \{(0,y,z)\}$, then $V = W_1 \oplus W_2$.