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Question on change of basis

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    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
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    ReputationRep:
    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...
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    (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 \{ 1,x,x^2\} to your new basis S=\{e_1,e_2,e_3\}=\{ 1+x, 2+x^2, x-x^2\}

    Suppose q(x)=ae_1+be_2+ce_3, then you can set up three simultaneous equations:

    a+2b=2, a+c=3, b-c=4 and solve them

    You should find that q(x)=12e_1-5e_2-9e_3

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