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これが一次独立ということを示してください。

To show that the vectors \( x + 1 \), \( x^2 - 1 \), and \( x^3 + 1 \) are linearly independent, we need to verify that the only solution to the equation

\[ a(x + 1) + b(x^2 - 1) + c(x^3 + 1) = 0 \]

is \( a = 0 \), \( b = 0 \), and \( c = 0 \).

1. First, expand the given equation:
\[ a(x + 1) + b(x^2 - 1) + c(x^3 + 1) = 0 \]
\[ ax + a + bx^2 - b + cx^3 + c = 0 \]

2. Combine like terms:
\[ cx^3 + bx^2 + ax + (a - b + c) = 0 \]

3. For the above polynomial to be identically zero, the coefficients of each power of \( x \) must be zero. This gives us the following system of equations:
\[
\begin{cases}
c = 0 \quad \text{(coefficient of } x^3) \\
b = 0 \quad \text{(coefficient of } x^2) \\
a = 0 \quad \text{(coefficient of } x) \\
a - b + c = 0 \quad \text{(constant term)}
\end{cases}
\]

4. From the first three equations, we get \( a = 0 \), \( b = 0 \), and \( c = 0 \). Substituting these into the fourth equation, we confirm it holds true:
\[ 0 - 0 + 0 = 0 \]

Since the only solution is \( a = 0 \), \( b = 0 \), and \( c = 0 \), the vectors \( x + 1 \), \( x^2 - 1 \), and \( x^3 + 1 \) are linearly independent.

Would you like any additional details or have any questions?

Here are some further questions you might be interested in:
1. How can we determine if a larger set of vectors is linearly independent?
2. What is the geometric interpretation of linear independence in three dimensions?
3. How do we find the basis of a vector space?
4. What is the rank of a matrix, and how is it related to linear independence?
5. Can you explain the concept of a null space and its significance in linear algebra?

**Tip:** When checking for linear independence, it's useful to set up and solve the corresponding system of equations to ensure that the only solution is the trivial one.

Discover how to prove the linear independence of the vectors x + 1, x^2 - 1, and x^3 + 1 with a comprehensive step-by-step guide. This problem illustrates key concepts in linear algebra, focusing on vector spaces and independence. Ideal for students in Grades 10-12.

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