Show that the functions [tex](c_{1}+c_{2}sin^{2}x+c_{3}cos^2{x})[/tex] form a vector space. Find a basis of it. What is its dimension?(adsbygoogle = window.adsbygoogle || []).push({});

My answer is that it's a vector space because:

[tex](c_{1}+c_{2}sin^{2}x+c_{3}cos^2{x})+(c'_{1}+c'_{2}sin^{2}x+c'_{3}cos^2{x})

=(c_{1}+c'_{1}+(c_{2}+c'_{2})sin^{2}x+(c_{3}+c'_{3})cos^2{x})[/tex] which is a function in the same form as the original function.

Basically all combinations of sums of multiples create functions of the same form as the original function. It's dimension is two because it's of one variable, x.

However, I not sure about the basis. I want to say it's just the original function, but I don't know why. I'm a little rusty when it comes to basis stuff.

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# Homework Help: Show functions of this form are a vector space etc

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