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I am trying to show that the vector space $V=span\{f_1,f_2\}$, $f_1=\sin(x)$, $f_2=\cos(x)$ has a basis

$$\underline{g}=(g_1,g_2)$$$$ g_1=2\sin(x)+\cos(x), g_2=3\cos(x)$$

What I've tried:

I manage to show that the $g_1$ and $g_2$ are lineary independent. The thing I have problem with is to show that $g_1$ and $g_2$ span $V$. I'm trying to use the definition of span to see if I can express $f_1$ and $f_2$ as a linear combination of $g_1$ and $g_2$, however, I'm not sure how to fulfill this proof on paper.

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Let $h$ be an element in $V$, then $h(x) = a\sin(x) + b\cos(x)$ for some real numbers $a$ and $b$. So we need to show that we can find numbers $c$, and $d$ so that

$$a\sin(x) + b\cos(x) = c(2\sin(x) + \cos(x)) + d(3\cos(x)).$$

This amounts to showing that $2c = a$ and $c + 3d = b$.

So choose $c = a/2$ and $d = (b - c)/3 = b/3 - a/6$ and we're done.

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  • $\begingroup$ Thank you! I was onto that type of comparison, but I was not sure if that completed the proof. $\endgroup$
    – Curtain
    Feb 2, 2014 at 8:37

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