The problem defines $V = P_{2024}(\mathbb{R})$, the vector space of all polynomials of degree at most 2024 with real coefficients. It further defines two subspaces $W$ and $U$ as follows:

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

We are tasked to find the dimensions of the subspaces $W \cap U$ and $W + U$.

Step 1: Dimension of $V$

The dimension of $V$, the vector space of all polynomials of degree at most 2024, is: $\dim(V) = 2025.$

Step 2: Understanding $W$

The subspace $W$ consists of polynomials in $V$ that satisfy: $p(0) = 0 \quad \text{and} \quad p(1) = 0.$ This means $p(x)$ must have roots at $x = 0$ and $x = 1$, so $p(x)$ can be written as: $p(x) = x(x-1)q(x),$ where $q(x)$ is any polynomial of degree at most $2022$. Therefore: $\dim(W) = 2023.$

Step 3: Understanding $U$

The subspace $U$ consists of polynomials in $V$ that satisfy: $p(0) = 0 \quad \text{and} \quad p(-1) = 0.$ This means $p(x)$ must have roots at $x = 0$ and $x = -1$, so $p(x)$ can be written as: $p(x) = x(x+1)q(x),$ where $q(x)$ is any polynomial of degree at most $2022$. Therefore: $\dim(U) = 2023.$

Step 4: Understanding $W \cap U$

The subspace $W \cap U$ consists of polynomials in $V$ that satisfy: $p(0) = 0, \quad p(1) = 0, \quad \text{and} \quad p(-1) = 0.$ This means $p(x)$ must have roots at $x = 0$, $x = 1$, and $x = -1$, so $p(x)$ can be written as: $p(x) = x(x-1)(x+1)q(x),$ where $q(x)$ is any polynomial of degree at most $2021$. Therefore: $\dim(W \cap U) = 2022.$

Step 5: Understanding $W + U$

Using the dimension formula for the sum of two subspaces: $\dim(W + U) = \dim(W) + \dim(U) - \dim(W \cap U).$ Substituting the known dimensions: $\dim(W + U) = 2023 + 2023 - 2022 = 2024.$

Final Answers

1. $\dim(W \cap U) = 2022$,
2. $\dim(W + U) = 2024$.

Would you like a detailed explanation of any specific step? Let me know!

Relative Questions:

1. What happens if we consider $V = P_n(\mathbb{R})$ for a general $n$ instead of 2024?
2. How does the dimension formula for $W + U$ apply to other vector spaces?
3. Why does $p(0) = 0$ imply that $p(x)$ must be divisible by $x$?
4. Can we use similar methods to compute dimensions in finite fields instead of real numbers?
5. How do properties of polynomial roots relate to vector space subspaces?

Tip:

Always use the dimension formula for the sum of subspaces to verify results in vector space problems!