show that {1 + x + x^2, x + x^2, 1 + x^2} spans P2

G

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Show that {1 + x + x^2, x + x^2, 1 + x^2} spans P2

The way I did this was by putting this into a matrix and getting Reduced Echelon Form:
This is a matrix below:

[1 0 1]
[1 1 0]
[1 1 1]

The question I have is if I'm supposed to put the vectors into rows and then get reduced echelon form like this:
[1 1 1]
[0 1 1]
[1 0 1]

But it seems to me that they are already in rows, so I'm wondering which way is correct.

Thanks
Take care,
Beckie[/tex]
 
\(\displaystyle \L\\a_{0}+a_{1}x+a_{2}x^{2}=ap_{1}+bp_{2}+cp_{3}\)

Equate coefficients:

\(\displaystyle a_{0}=a+c\\a_{1}=a+b\\a_{2}=a+b+c\)

\(\displaystyle a_{0}=1\ 0\ 1\\a_{1}=1\ 1\ 0\\a_{2}=1\ 1\ 1\)

rref:

\(\displaystyle 1\ 0\ 0\ a_{0}+a_{1}-a_{2}\\0\ 1\ 0\ a_{2}-a_{0}\\0\ 0\ 1\ a_{2}-a_{1}\)


Does it span?.
 
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