Question:

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: Dimension of $V$

The space $V = P_{2024}(\mathbb{R})$ consists of all polynomials of degree at most 2024. Thus, the dimension of $V$ is:

$\dim(V) = 2025.$

Step 2: Dimension of $W$

A polynomial $p(x) \in W$ satisfies $p(0) = 0$ and $p(1) = 0$. These two conditions are linear constraints on the coefficients of $p(x)$. Imposing two conditions reduces the dimension of $V$ by 2, so:

$\dim(W) = \dim(V) - 2 = 2025 - 2 = 2023.$

Step 3: Dimension of $U$

A polynomial $p(x) \in U$ satisfies $p(0) = 0$ and $p(-1) = 0$. These are also two linear conditions, so:

$\dim(U) = \dim(V) - 2 = 2025 - 2 = 2023.$

Step 4: Dimension of $W \cap U$

A polynomial $p(x) \in W \cap U$ satisfies the conditions:

1. $p(0) = 0$,
2. $p(1) = 0$,
3. $p(-1) = 0$.

These are three independent linear constraints. Therefore, the dimension of $W \cap U$ is:

$\dim(W \cap U) = \dim(V) - 3 = 2025 - 3 = 2022.$

Step 5: Dimension of $W + U$

Using the formula for the dimension of the sum of two subspaces:

$\dim(W + U) = \dim(W) + \dim(U) - \dim(W \cap U).$

Substitute the known dimensions:

$\dim(W + U) = 2023 + 2023 - 2022 = 2024.$

Final Answer:

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

Would you like a detailed explanation of any step? Here are some related questions for further exploration:

1. How do we determine the independence of constraints in polynomial vector spaces?
2. What is the general formula for the dimension of $W \cap U$ and $W + U$ in terms of $\dim(W)$, $\dim(U)$, and $\dim(V)$?
3. Can this problem be extended to polynomials of arbitrary degree $n$?
4. How do changes in the constraints $p(1) = 0$ and $p(-1) = 0$ affect the solution?
5. What happens if $W$ and $U$ are defined with different roots instead of $0, 1,$ and $-1$?

Tip:

Always confirm the independence of constraints before subtracting dimensions in polynomial spaces.