Hello,
I am asked this:
Let B = {1 + x, x + x^2, 1 + x^2} be an ordered basis of P2.
And let B' be the standard basis of P2.
Find the coordinate vector of p(x) = 1 + 2x - x^2 with respect to B.
Any help would be greatly appreciated, I am not even sure how to approach this.
Thank you so much.
I am asked this:
Let B = {1 + x, x + x^2, 1 + x^2} be an ordered basis of P2.
And let B' be the standard basis of P2.
Find the coordinate vector of p(x) = 1 + 2x - x^2 with respect to B.
Any help would be greatly appreciated, I am not even sure how to approach this.
Thank you so much.
-
If the coordinate vector for p relative to B is (a, b, c), then
p(x) = a(1 + x) + b(x + x²) + c(1 + x²).
So you have a linear system
1 + 2x - x² = (a + c) + (a + b)x + (b + c)x² ==>
a + c = 1
a + b = 2
b + c = -1 ==> (a, b, c) = (2, 0, -1)
Perhaps you can see from this that (a, b, c) is the product of the matrix A^(-1) with the coordinate vector for p with respect to the elementary basis where A is the matrix whose columns are the coordinate vectors for the basis elements (in the order given) with respect to the elementary basis.
p(x) = a(1 + x) + b(x + x²) + c(1 + x²).
So you have a linear system
1 + 2x - x² = (a + c) + (a + b)x + (b + c)x² ==>
a + c = 1
a + b = 2
b + c = -1 ==> (a, b, c) = (2, 0, -1)
Perhaps you can see from this that (a, b, c) is the product of the matrix A^(-1) with the coordinate vector for p with respect to the elementary basis where A is the matrix whose columns are the coordinate vectors for the basis elements (in the order given) with respect to the elementary basis.