The set 𝐵={𝑥2−3, 𝑥−6+2𝑥2, 3𝑥−19+7𝑥2} is a basis...

Question

 The set 𝐵={𝑥2−3, 𝑥−6+2𝑥2, 3𝑥−19+7𝑥2} is a basis for P₂. Find the coordinates of 𝑝(𝑥)=82−11𝑥−30𝑥2 relative to this basis

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[𝑝(𝑥)]𝐵= | _ |

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Answer 1

Let coordinates of P(x) = <a, b,c>

Then 82 - 11x - 30x² = a(x² - 3) + b(x - 6 + 2x²) + c(3x - 19 + 7x²)

We get system of equations:

a + 2b + 7c = - 30 (1) (1) - 2(2) a + c = - 8 (4) (4) + (5): - 2a = 8; a = - 4, c = - 8 + 4 = - 4

b + 3c = - 11 (2) (3) + 6(2) - 3a - c = 16 (5) From (2) b = - 11 - 3(- 4) = - 11 + 12 = 1

-3a - 6b - 19c = 82 (3)

So, vector p(x) = <- 4, 1, - 4>