The question reads as follows:

Let $V = P_{2024}(\mathbb{R})$, the vector space of all polynomials with real coefficients of degree at most 2024. Let $W$ and $U$ be two subspaces of $V$ described by the following:

• $W = \{ p(x) \in V : p(0) = p(1) = 0 \}$
• $U = \{ p(x) \in V : p(0) = p(-1) = 0 \}$

Find the dimensions of the subspaces $W \cap U$ and $W + U$.

Solution

1. Dimension of $V$: The vector space $P_{2024}(\mathbb{R})$ consists of all polynomials of degree at most 2024. The dimension of $V$ is: $\dim(V) = 2025$

2. Dimension of $W$: Subspace $W$ consists of polynomials such that $p(0) = 0$ and $p(1) = 0$. These conditions impose two constraints on the coefficients of a general polynomial. Hence: $\dim(W) = 2025 - 2 = 2023$

3. Dimension of $U$: Similarly, $U$ consists of polynomials such that $p(0) = 0$ and $p(-1) = 0$. These are also two constraints, so: $\dim(U) = 2025 - 2 = 2023$

4. Intersection $W \cap U$: $W \cap U$ consists of polynomials satisfying all the constraints of both $W$ and $U$, namely: $p(0) = 0, \quad p(1) = 0, \quad \text{and} \quad p(-1) = 0$ These impose three independent constraints, so: $\dim(W \cap U) = 2025 - 3 = 2022$

5. Sum $W + U$: Using the formula for the dimension of a sum of subspaces: $\dim(W + U) = \dim(W) + \dim(U) - \dim(W \cap U)$ Substituting the values: $\dim(W + U) = 2023 + 2023 - 2022 = 2024$

Final Answers:

• $\dim(W \cap U) = 2022$
• $\dim(W + U) = 2024$

Would you like a step-by-step explanation or further clarification?

Related Questions:

1. How do we determine the general formula for the intersection of subspaces in polynomial vector spaces?
2. What does the dimension of $W + U$ signify geometrically in this context?
3. How would the solution change if the vector space were $P_{n}(\mathbb{R})$ with $n < 2024$?
4. Can we generalize these results to polynomial spaces over other fields (e.g., $\mathbb{C}$ or $\mathbb{Q}$)?
5. How does the rank-nullity theorem connect to these types of problems?

Tip: Always count the constraints carefully when determining the dimension of subspaces, as errors often stem from missing dependent relationships among constraints.