Given $U = \{ p \in P_4(\mathbb{R}) : p''(6) = 0 \}$, we need to find a basis for $U$. Here's the step-by-step breakdown:

Step 1: General form of a polynomial in $P_4(\mathbb{R})$

A polynomial in $P_4(\mathbb{R})$ is of degree at most 4, so we can express a general element of $P_4(\mathbb{R})$ as: $p(x) = ax^4 + bx^3 + cx^2 + dx + e$ where $a, b, c, d, e \in \mathbb{R}$ are constants.

Step 2: Second derivative of $p(x)$

To impose the condition $p''(6) = 0$, we first compute the second derivative of $p(x)$.

First derivative: $p'(x) = 4ax^3 + 3bx^2 + 2cx + d$

Second derivative: $p''(x) = 12ax^2 + 6bx + 2c$

Step 3: Apply the condition $p''(6) = 0$

Substitute $x = 6$ into the second derivative and set it equal to 0: $12a(6)^2 + 6b(6) + 2c = 0$ Simplifying: $432a + 36b + 2c = 0$ Dividing the entire equation by 2: $216a + 18b + c = 0$ Thus, the condition that must hold for $p(x) \in U$ is: $c = -216a - 18b$

Step 4: Express $p(x)$ in terms of free variables

From the condition above, we can express $c$ in terms of $a$ and $b$. So, the polynomial $p(x)$ becomes: $p(x) = ax^4 + bx^3 + (-216a - 18b)x^2 + dx + e$

Step 5: Construct a basis

The free variables are $a, b, d, e$, so we can generate a basis by assigning one of these variables to 1 and the others to 0 in turn:

1. Set $a = 1$, $b = 0$, $d = 0$, $e = 0$: $p(x) = x^4 - 216x^2$

2. Set $a = 0$, $b = 1$, $d = 0$, $e = 0$: $p(x) = x^3 - 18x^2$

3. Set $a = 0$, $b = 0$, $d = 1$, $e = 0$: $p(x) = x$

4. Set $a = 0$, $b = 0$, $d = 0$, $e = 1$: $p(x) = 1$

Step 6: Final basis

Thus, a basis for $U$ is: $\{ x^4 - 216x^2, x^3 - 18x^2, x, 1 \}$

Would you like more details on any part of the solution or have any questions?

Here are 5 related questions to consider:

1. What is the dimension of the space $P_4(\mathbb{R})$?
2. How would the basis change if the condition was $p'(6) = 0$ instead of $p''(6) = 0$?
3. What does it mean geometrically for a polynomial to satisfy $p''(6) = 0$?
4. How would you find a basis if the degree of the polynomial was higher, say $P_5(\mathbb{R})$?
5. How can this method be extended to polynomials over other fields?

Tip: When solving for a subspace of polynomials, always focus on the conditions imposed on the derivatives or values to reduce the degrees of freedom.