LINEAR ALGEBRA Chapter 4 Section 42 Question 10b Express the

LINEAR ALGEBRA

Chapter 4, Section 4.2, Question 10b Express the following as a linear combination of P1 5 x 6x2, p2 5 -x 5x2, and p3- 7 3x 3x2. Click here to enter or edit your answer 61 5x 51x

Solution

simply express given vector p as a linear combination of the vectors p1, p2, p3 with unknown coefficients:

p = a1*p1 + a2*p2 + a3*p3

-9 -7x - 15x² = a1(2 + x + 4x²) + a2(1 - x + 3x²) + a3(3 + 2x + 5x²)

-9 -7x - 15x² = 2a1 + a1*x + 4a1*x² + a2 - a2*x + 3a2*x² + 3a3 + 2a3*x + 5a3*x²

-9 -7x - 15x² = (2a1 + a2 + 3a3) + x(a1 - a2 + 2a3) + x²(4a1 + 3a2 + 5a3)

Equate coefficients on the both sides by 1, x and x²:

2a1 + a2 + 3a3 = -9
a1 - a2 + 2a3 = -7
4a1 + 3a2 + 5a3 = -15