Let's go through each part of the problem and solve it in detail.

Part (a)

We are asked to determine the coordinates of the vector $(0, 6, 7, 8)$ in $\mathbb{R}^4$ with respect to the basis $\{(1, 1, 1, 1), (0, 1, 1, 1), (0, 0, 1, 1), (0, 0, 0, 1)\}$.

1. Let the coordinates of $(0, 6, 7, 8)$ with respect to this basis be $(a, b, c, d)$.
2. Then, we can write: $a(1, 1, 1, 1) + b(0, 1, 1, 1) + c(0, 0, 1, 1) + d(0, 0, 0, 1) = (0, 6, 7, 8).$
3. Expanding the left-hand side, we get: $(a, a + b, a + b + c, a + b + c + d) = (0, 6, 7, 8).$
4. Now we can set up equations by comparing components:
   • $a = 0$
   • $a + b = 6$ → $b = 6$
   • $a + b + c = 7$ → $c = 1$
   • $a + b + c + d = 8$ → $d = 1$

Therefore, the coordinates of $(0, 6, 7, 8)$ in terms of the given basis are $(a, b, c, d) = (0, 6, 1, 1)$.

Part (b)

We are given a polynomial space $P_3(\mathbb{R})$ with basis $S = \{1, t, t^2, t^3\}$. We are asked to find the coordinates of the polynomial $p(t) = 6 + 7t^2 + 8t^3$ with respect to this basis.

1. Write $p(t)$ as a linear combination of the basis elements: $p(t) = a \cdot 1 + b \cdot t + c \cdot t^2 + d \cdot t^3.$
2. Equate coefficients to match $p(t) = 6 + 7t^2 + 8t^3$:
   • The constant term gives $a = 6$.
   • The coefficient of $t$ is zero, so $b = 0$.
   • The coefficient of $t^2$ gives $c = 7$.
   • The coefficient of $t^3$ gives $d = 8$.

Thus, the coordinates of $p(t)$ with respect to the basis $S$ are $(a, b, c, d) = (6, 0, 7, 8)$.

Let me know if you need further details or explanations on any part of the solution.

Here are some follow-up questions related to this topic:

1. What is the general method for finding the coordinates of a vector with respect to a given basis in any vector space?
2. How can we verify that a given set of vectors forms a basis for a vector space?
3. How does one determine if a polynomial space $P_n(\mathbb{R})$ is a vector space?
4. What are the applications of finding coordinates with respect to different bases in linear algebra?
5. How would the approach change if the vector space were $\mathbb{R}^n$ with an arbitrary $n$?

Tip: When finding coordinates with respect to a basis, setting up equations for each component by matching terms is an effective method.