$Linear independence of functions$

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$Linear independence of functions$

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(i) the functions $f_1(t)=t e^t $ and $f_2(t)=e^t$ are linearly independent on $(0,1)$.
(ii) $f_1(t)=t e^t $ and $f_2(t)=e^t$ are linearly dependent for every fixed $t\in (0,1)$.

I confused about this question. Part (i) and (ii) are really similar, but they are asking me two prove completely different things.

Linear Algebra

Elviegem

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Answer 1

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(i) Suppose
\[c_1 te^t+c_2e^t=0,\]
then \[(1) c_1t+c_2=0\]
for every $t \in (0,1)$. Since (1) holds for all $t\in (0,1)$, taking a derivative from both sides we get
\[c_1=0,\]
and substituting in (1) we also get $c_2=0$. Hence $c_1=c_2=0$, and hence (be definition) $f_1$ and $f_2$ are linearly independent.

(ii) Now fix $t \in (0,1)$. Then note that $t$ is a constant and not a variable anymore. Then

\[f_1=te^{t}=tf_2.\]
Hence for any fixed $t=t_0$ we have
\[f_1=Cf_2,\]
where $C=t_0$ is a constant. So $f_1$ and $f_2$ are linearly dependent for every fixed $t$.

The difference between part (i) and part (ii) is that in part (i) the question asks for linear independence of two functions defined on (0,1), but in part (ii) the problem asks for linear independence of two sets of numbers parametrized by some variable. So they are fundamentally different questions.

Erdos

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$Linear independence of functions$

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