Final answer:
To write -1-9x-x² as a linear combination of elements from the basis C={2, 2-3x, -3-3x-x²}, we can set up a system of equations and solve for the coefficients. The coefficients are k₁ = -1/2, k₂ = -3/2, and k₃ = 1/2. Therefore, the linear combination of elements is -1-9x-x² = (-1/2)(2) + (-3/2)(2-3x) + (1/2)(-3-3x-x²).
Step-by-step explanation:
To write -1-9x-x² as a linear combination of elements from the basis C={2, 2-3x, -3-3x-x²}, we need to find coefficients k₁, k₂, and k₃ such that: -1-9x-x² = k₁(2) + k₂(2-3x) + k₃(-3-3x-x²).
Expanding the right side of the equation, we get: -1-9x-x² = 2k₁ + (2k₂ - 3k₃)x + (-3k₃ - k₃)x².
By comparing the coefficients, we can set up a system of equations:
2k₁ = -1,
2k₂ - 3k₃ = -9,
-3k₃ - k₃ = -1.
Solving this system of equations, we find that k₁ = -1/2, k₂ = -3/2, and k₃ = 1/2. Therefore, the linear combination of elements from basis C that equals -1-9x-x² is: -1-9x-x² = (-1/2)(2) + (-3/2)(2-3x) + (1/2)(-3-3x-x²).