Verify linear independence of the following sets of vectors: $\{1−x, x+x^2, 1+ x^2 \} $ in $\Bbb R[x]$ over $\Bbb R$
How to solve this?
We are checking $a(1-x)+b(x+x^2)+c(1+x^2)=0$, but it gives $x^2=-1$.
I don't know how to continue.
Verify Linear independence
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linear-algebra
1 Answers
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$$a(1-x)+b(x+x^2)+c(1+x^2)=0 \to (b+c)x^2+(b-a)x+(a+c)=0$$
We have then a polynomial identity, because the above equation has to be true for every $x \in \Bbb R$, then:
$$b+c=0\\ b-a=0\\ a+c=0$$
what give us $a=b$ and $c=-b$.
That give us infinite non null solutions (just take $b\ne 0$) and then they are linearly dependent.