Algebra. Lost! Coordinate vector. Basis.
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Algebra. Lost! Coordinate vector. Basis.

[From: ] [author: ] [Date: 13-07-04] [Hit: ]
b, c),p(x) = a(1 + x) + b(x + x²) + c(1 + x²).b + c = -1 ==> (a, b, c) = (2,......
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.

-
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.
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