Linearly independent sets

**chella182** · December 11th 2009, 02:20 AM

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

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

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

**HallsofIvy** · December 11th 2009, 04:46 AM

Originally Posted by chella182

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

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

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 $a_1f_1(x)+ a_2f_2(x)+ \cdot\cdot\cdot+ a_nf_n(x)= 0$ , for all x, is to have $a_1= a_2= \cdot\cdot\cdot= a_n= 0$ .

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

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

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

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

**chella182** · December 11th 2009, 06:59 AM

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.

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