Final answer:
To determine if x - x³ is in the span of x, x², and x² - x³, we need to check if x - x³ can be written as a linear combination of x, x², and x² - x³.
Step-by-step explanation:
To determine if x - x³ is in the span of x, x², and x² - x³, we need to check if x - x³ can be written as a linear combination of x, x², and x² - x³. In other words, we want to find coefficients a, b, and c such that:
- ax + bx² + c(x² - x³) = x - x³
By expanding and simplifying this equation, we get:
- (a - c)x³ + (b - c)x² + (a - b)x = 0
This equation must hold for all values of x. Therefore, the coefficients a - c, b - c, and a - b must all be equal to zero. This leads to the following system of equations:
- a - c = 0
- b - c = 0
- a - b = 0
Solving this system, we find that a = b = c. Therefore, x - x³ is in the span of x, x², and x² - x³.