Let's break down the problem step by step.

Part 1:

You are given a basis for the vector space of all cubic polynomials: $\{1 + x, 1 + x^2, 1 - x + x^2, x + x^3\}$

This means that any cubic polynomial can be written as a linear combination of these four polynomials.

Part 2:

You are asked to find the coordinate of $1 + x$ with respect to this basis. That is, you need to express $1 + x$ as a linear combination of the given basis vectors: $1 + x = c_1(1 + x) + c_2(1 + x^2) + c_3(1 - x + x^2) + c_4(x + x^3)$ Where $c_1, c_2, c_3, c_4$ are scalars (coordinates) to be determined.

Step-by-Step Process:

1. Express the equation: We already have the polynomial on the left-hand side, $1 + x$. Now, we expand the right-hand side: $c_1(1 + x) + c_2(1 + x^2) + c_3(1 - x + x^2) + c_4(x + x^3)$ Expanding each term: $c_1(1 + x) = c_1 + c_1x$ $c_2(1 + x^2) = c_2 + c_2x^2$ $c_3(1 - x + x^2) = c_3 - c_3x + c_3x^2$ $c_4(x + x^3) = c_4x + c_4x^3$

   Adding these together: $(c_1 + c_2 + c_3) + (c_1 - c_3 + c_4)x + (c_2 + c_3)x^2 + c_4x^3$

2. Match the coefficients: Now, match the coefficients of $1$, $x$, $x^2$, and $x^3$ from both sides:

   • Constant term: $1 = c_1 + c_2 + c_3$
   • Coefficient of $x$: $1 = c_1 - c_3 + c_4$
   • Coefficient of $x^2$: $0 = c_2 + c_3$
   • Coefficient of $x^3$: $0 = c_4$

3. Solve the system: From the equation $c_4 = 0$, we substitute into the other equations:

   • $0 = c_4$, so the equation for $x^3$ is already satisfied.
   • From $c_2 + c_3 = 0$, we get $c_3 = -c_2$.
   • Substituting $c_3 = -c_2$ into the equation $c_1 - c_3 + c_4 = 1$, we get: $c_1 + c_2 = 1$
   • Now substitute $c_3 = -c_2$ into $c_1 + c_2 + c_3 = 1$, which simplifies to: $c_1 = 1$

4. Find the remaining coefficients:

   • We now know $c_1 = 1$. Substituting this into $c_1 + c_2 = 1$, we find $c_2 = 0$.
   • Since $c_3 = -c_2$, we get $c_3 = 0$.
   • Finally, we already know $c_4 = 0$.

Conclusion:

The coordinates of $1 + x$ with respect to the given basis are: $[1 + x] = (1, 0, 0, 0)$

Would you like further details on any of these steps?

Five Related Questions:

1. What does it mean for a set of polynomials to form a basis for a vector space?
2. How would you find the coordinate vector of a higher-degree polynomial with respect to a basis?
3. Can you always express any polynomial as a linear combination of basis polynomials?
4. What is the dimension of the vector space of cubic polynomials?
5. How does the concept of polynomial vector spaces extend to multivariable polynomials?

Tip:

When finding coordinates with respect to a basis, ensure the system of equations you form from comparing coefficients is consistent and reflects each power of $x$.