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Question on change of basis

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1. I am wondering how to write q(x)= 2 + 3x + 4x2

in terms of the basis S={1+x, 2+x2, x-x2}, without using 'observation'. I remember there was a matrix method.

Any help would be great
2. S is a basis you don't like
Can you write a more preferable basis in terms of S? i.e. 1,x, xx
This is a basis change...
3. (Original post by Lewk)
I am wondering how to write q(x)= 2 + 3x + 4x2

in terms of the basis S={1+x, 2+x2, x-x2}, without using 'observation'. I remember there was a matrix method.

Any help would be great
This is essentially a change from the basis to your new basis

Suppose , then you can set up three simultaneous equations:

and solve them

You should find that

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