# Thread: Linearly independent sets

1. ## Linearly independent sets

Are the following sets of functions linearly independent or not? Explain your answer.

a) $\displaystyle f(X):= \sin{(2X)}+e^X,f'(X),f''(X),f'''(X)$.
b) $\displaystyle f(X):=X^3+1,f(X)^2,f(X)^3$
c) $\displaystyle \cos{(2x)},\sin{(X)}^2,\cos{(X)}^2$

Literally no clue here, our teacher for this module is shockingly bad. I would ask you not to be vague in your answers, but I'm asking for help it would be a little rude, but yeah... assume I know little to nothing about this topic

2. Originally Posted by chella182
Are the following sets of functions linearly independent or not? Explain your answer.

a) $\displaystyle f(X):= \sin{(2X)}+e^X,f'(X),f''(X),f'''(X)$.
b) $\displaystyle f(X):=X^3+1,f(X)^2,f(X)^3$
c) $\displaystyle \cos{(2x)},\sin{(X)}^2,\cos{(X)}^2$

Literally no clue here, our teacher for this module is shockingly bad. I would ask you not to be vague in your answers, but I'm asking for help it would be a little rude, but yeah... assume I know little to nothing about this topic
Blaming your teacher isn't going to help you at all. No matter how how bad your teacher is, you surely should be able to look up the definition of "linearly independent". For functions it is that the only way you can have $\displaystyle a_1f_1(x)+ a_2f_2(x)+ \cdot\cdot\cdot+ a_nf_n(x)= 0$, for all x, is to have $\displaystyle a_1= a_2= \cdot\cdot\cdot= a_n= 0$.

a) If $\displaystyle f(x)= sin(2x)+ e^x$, what are f'(x), f"(x), and f'''(x)? Suppose $\displaystyle a_1f(x)+ a_2f'(x)+ a_3f"(x)+ a_4f'''(x)= 0$ for all x. Taking four different values for x will give you four linear equations to solve for $\displaystyle a_1$, $\displaystyle a_2$, $\displaystyle a_3$, and $\displaystyle a_4$.

b) If $\displaystyle f(x)= x^3+ 1$, what are $\displaystyle f^2$ and $\displaystyle f^3$. Suppose $\displaystyle a_1f(x)+ a_2f^2(x)+ a_3f^3(x)= 0$ for all x. Taking three different values for x will give you three linear equations to solve.

c) I assume that by "$\displaystyle cos(x)^2$ you mean $\displaystyle cos^2(x)$, not $\displaystyle cos(x^2)$.

Again, suppose you have $\displaystyle a_1cos(2x)+ a_2sin^2(x)+ a_3cos^2(x)= 0$. Taking three different values for x will give you three linear equations to solve.

3. Originally Posted by HallsofIvy
Blaming your teacher isn't going to help you at all. No matter how how bad your teacher is, you surely should be able to look up the definition of "linearly independent".
Ok. Thanks for the help and all, I do kind of understand what you're saying which should help with the work, but you are working the rather rude and risky assumption that I haven't tried to look things up for myself. I assure you this website is always my last resort.