https://file.aiquickdraw.com/mathbot/file/17278794981726yb1d9fj.jpg

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 \(x\).

The polynomials 1 + x, 1 + x^2, 1 − x + x^2, x + x^3 form a basis for all cubic polynomials. Find the coordinate of 1 + x with respect to the basis from the previous problem.

Learn how to find the coordinate of the polynomial 1 + x with respect to a given basis for cubic polynomials. This problem covers linear combinations, polynomial basis, and vector spaces. Suitable for undergraduate-level students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation