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Question on change of basis

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 :smile:
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jj193
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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...
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 :smile:


This is essentially a change from the basis {1,x,x2}\{ 1,x,x^2\} to your new basis S={e1,e2,e3}={1+x,2+x2,x−x2}S=\{e_1,e_2,e_3\}=\{ 1+x, 2+x^2, x-x^2\}

Suppose q(x)=ae1+be2+ce3q(x)=ae_1+be_2+ce_3, then you can set up three simultaneous equations:

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

You should find that q(x)=12e1−5e2−9e3q(x)=12e_1-5e_2-9e_3

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