Hi guys, this is a linear algebra question. Let $$V \subseteq Map(\mathbb{R},\mathbb{R}) $$ be the sub-vector space of continuous functions. Let W be the sub-vector space of V spanned by the functions 1, sin x, and sin^2 x.

Show that: $$1, \sin(x), \sin(x^2) \in W $$ are linearly independent.

I have no idea how to approach this... the only thing I know about linear independence is in relation to vectors. Not to functions.

Question

Hi guys, this is a linear algebra question. Let $$V \subseteq Map(\mathbb{R},\mathbb{R}) $$ be the sub-vector space of continuous functions. Let W be the sub-vector space of V spanned by the functions 1, sin x, and sin^2 x. 

Show that: $$1, \sin(x), \sin(x^2) \in W $$ are linearly independent.

I have no idea how to approach this... the only thing I know about linear independence is in relation to vectors. Not to functions.

anonymous · Answer

you have to show that if a linear combination is zero, then each coefficient must be zero

anonymous · Answer

in other words is 
$$a_1+a_2\sin(x)+a_3\sin^2(x)=0$$ then 
$$a_1=a_2=a_3=0$$

anonymous · Answer

in this context, the meaning of $$a_1+a_2\sin(x)+a_3\sin^2(x)=0$$is that it is the zero function, i.e. it is zero for all values of $x$ not just for some value of $x$

anonymous · Answer

at the moment, however, i totally forget how to do this, but there should be some similar example in a text.  i believe you can do it by the "wronskian"

loser66 · Answer

:) i have a question, sin^2x or sin (x^2) ?

anonymous · Answer

$$ f(g'h'' - h'g'') - g(f 'h'' - h'f '') +h(f 'g'' - g'f'')$$

anonymous · Answer

here is an example http://www.youtube.com/watch?v=snbmY5oQMlQ

anonymous · Answer

Sorry, took me a while to figure out how to make a pciture of it.