In space of real functions we have subspace $\textbf{V}$ dedicated by set $V = \langle 1,x,\cos x,\sin x \rangle$ and its subspace $\textbf{W}$ dedicated by set $W = \lbrace f \in V : f(\pi)=0\rbrace$.
I need to find base of space $\textbf{W}$
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$\begingroup$ You are pretty new to MathSE, but you should still know the following. You are more likely to get a good answer to your question if you follow a few guidelines. In particular, what have you tried so far, and just where are you stuck? This is not a homework-answering site: we want to see that you have put significant work into the problem. $\endgroup$– Rory DaultonNov 28, 2015 at 12:47
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$\begingroup$ this is not my homework, I'm not cheater. I'm just preparing for exam and I found this problem in textbook and now'm lost :-( $\endgroup$– MatFyzakNov 28, 2015 at 23:27
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$\begingroup$ You should make that clear in your main question. Also, as I said, tell us just where you are stuck. What part of this question can you do? You also have an excellent hint from Henning Makholm. What have you been able to do with it? $\endgroup$– Rory DaultonNov 29, 2015 at 0:25
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Hint: You already have a basis for $\mathbf V$ (though if you don't already know that $V$ is linearly independent, you'd need to show that before you can say for sure that $V$ is a basis for $\mathbf V$), so you can reduce the problems to working in that basis. You also know (hopefully) that $$ (a\cdot 1+bx+c\cos x+d\sin x)(\pi) = a+b\pi-c $$ So all you need to do is to find a basis for the solution space for the equation $$ a+\pi b-c+0d = 0 $$ and then map those basis vectors back to actual functions using your basis for $\mathbf V$.
answered Nov 28, 2015 at 10:59
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