$tf_1,t^2f_2,t^3f_3$ are linearly indipendent in $C[0,1]$ does this imply $f_1,f_2,f_3$ are linearly indipendent?

Question

Given that $tf_1,t^2f_2,t^3f_3$ are linearly indipendent in $C[0,1]$ does this imply $f_1,f_2,f_3$ are linearly indipendent? $t\to t^n$ is a polynomial, $f_i\in C[0,1]$

Thank you for helping.

Answer 1

I think this answer is no. $f_1=a,f_2=b,f_3=c$ and $a,b,c$ are constants.

$1,1,1$ are linear dependent on $C[0,1]$, however $t,t^2,t^3$ are linear independent.

Answer 2

No. Here is a counterexample: take $f_1, f_2, f_3$ to be non-zero constants. Then $f_1 t, f_2 t^2, f_3 t^3$ are certainly linearly independent, but the $f_j$, $1 \le j \le 3$, are certainly linearly dependent.

Hope this helps. Cheers,

and as always,

Fiat Lux!!!