The given problem is:

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

Step 1: Understanding $V$

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

Step 2: Subspace $W$

In $W$, the polynomials satisfy the conditions: $p(0) = 0 \quad \text{and} \quad p(1) = 0.$

These conditions imply two linear constraints on the coefficients of a general polynomial. Thus, the dimension of $W$ is: $\dim(W) = \dim(V) - 2 = 2025 - 2 = 2023.$

Step 3: Subspace $U$

In $U$, the polynomials satisfy the conditions: $p(0) = 0 \quad \text{and} \quad p(-1) = 0.$

Similarly, these two conditions impose two linear constraints. Hence: $\dim(U) = \dim(V) - 2 = 2025 - 2 = 2023.$

Step 4: Intersection $W \cap U$

The subspace $W \cap U$ consists of all polynomials satisfying the conditions: $p(0) = 0, \quad p(1) = 0, \quad \text{and} \quad p(-1) = 0.$

These three conditions are linearly independent and impose three constraints. Therefore: $\dim(W \cap U) = \dim(V) - 3 = 2025 - 3 = 2022.$

Step 5: Sum of Subspaces $W + U$

By the dimension formula for subspaces: $\dim(W + U) = \dim(W) + \dim(U) - \dim(W \cap U).$

Substituting the values: $\dim(W + U) = 2023 + 2023 - 2022 = 2024.$

Final Answer

1. The dimension of $W \cap U$ is: $\dim(W \cap U) = 2022.$

2. The dimension of $W + U$ is: $\dim(W + U) = 2024.$

Let me know if you'd like any further explanation or details. Here are five related questions to deepen understanding:

1. How does the dimension formula for subspaces help in solving similar problems?
2. What are other possible interpretations of $W \cap U$ and $W + U$?
3. How would the solution change if the degree of the polynomial were $n$ instead of 2024?
4. Can we generalize these results to non-polynomial vector spaces?
5. What if the conditions $p(0), p(1),$ and $p(-1)$ were dependent?

Tip: The dimension formula is a powerful tool for understanding relationships between subspaces. Use it systematically when working with sums and intersections!