The vectors P1X 5x2 X 2 P2x X2 3x 1 P3x 3x2 X 5 for

The vectors P_1(X) = 5x^2 + X + 2 P_2(x) = X^2 - 3x - 1 P_3(x) = 3x^2 - X + 5 form a basis for P_2 the vector space of polynomials of degree at most 2. Express the constant polynomial p(x) = 1 as a linear combination of p_1, p_2, P_3·

Solution

We have p₁(x) = 5x²+x+2, p₂(x) = x²-3x-1, and p₃(x) = 3x²-x+5. Let A =

5

1

3

0

1

-3

-1

0

2

-1

5

1

It may be observed that the columns of the above matrix comprise, the coefficients of p₁(x), p₂(x), p₃(x)and the constant polynomial 1.

We will reduce A to its RREf as under:

Multiply the 1st row by 1/5

Add -1 times the 1st row to the 2nd row

Add -2 times the 1st row to the 3rd row

Multiply the 2nd row by -5/6

Add 7/5 times the 2nd row to the 3rd row

Multiply the 3rd row by 3/17

Add -4/3 times the 3rd row to the 2nd row

Add -3/5 times the 3rd row to the 1st row

Add -1/5 times the 2nd row to the 1st row

Then, the RREF of A is

1

0

0

-1/17

0

1

0

-4/17

0

0

1

3/17

Hence 1 = -(1/17)p₁-(4/17)p₂ +(3/17)p₃

5	1	3	0
1	-3	-1	0
2	-1	5	1