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I am reading a book on partial differential equations. One of the exercise question in the book is:

Show that the functions $(c_1 + c_2 sin^2x + c_3 cos^2x)$ form a vector space. Find a basis of it. What is its dimension?

I don't know how important this question is for understanding PDE's, but I don't understand the question, nor do I know how to solve it. I know about standard vector spaces on the real numbers. Moreover, I've read about what abstract vector spaces are, but I think my abstract algebra is not developed enough to fully comprehend vector spaces over non-scalar fields.

So my question is: How can a "set of functions" form a vector space? how is that a meaningful statement? And how does this particular set of functions form a vector space?

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Take two functions which are in that particular form. Their "sum" (there is an obvious way to define sum of functions) is again in that same form. It forms an abelian group for this sum.

Also for any real number $k$ one can define $k$ times such function: it will gain be of the same form.

This sum defined on this set of functions satisfy all the properties that your "standard vector spaces on real numbers" satisfies. For example $f(x) = c_1+ c_2\sin^2 x + c_3\cos^2 x$, $g(x) = c_1' +c_2'\sin^2x+c_3'\cos^2x$, then $k(f(x)+ g(x)) = kf(x) + kg(x)$. Look at all the axioms of abstract vector spaces. All hold here. The scalars are still real numbers (or equally well complex numbers)

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  • $\begingroup$ So what does a "vector" in this vector space look like? is it simply $$\begin{pmatrix}{c_1\\c_2\\c_3}\end{pmatrix}$$? $\endgroup$ – user56834 Sep 27 '16 at 11:33
  • $\begingroup$ It is no more vector in the sense of geometry or physics. But as this system satisfies the same laws it is called vector space. However people rarely call individual elements as vectors. What you have written gives a way of identifying this vector space of functions with vector space of column vectors. You can check that in the same way polynomials of degree upto n having real coefficients form a vector space. $\endgroup$ – P Vanchinathan Sep 27 '16 at 11:52
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    $\begingroup$ @Programmer2134: The general element of this vector space looks like $(c_{1} + c_{2} \sin^{2} x + c_{3} \cos^{2} x)$. A vector isn't necessarily a column of numbers, but an element of a set equipped with operations of addition and scalar multiplication that satisfy several axioms. $\endgroup$ – Andrew D. Hwang Sep 27 '16 at 12:15
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A set of functions form a vector space if they obey the general definition of vector space.

To see that any set of vectors form a vector space, you can just check if the set has the following properties.

http://mathworld.wolfram.com/VectorSpace.html

As for the basis part of the question, you need to know something about linear independence. A basis is essentially the smallest linearly independent set that can span the set as a whole, i.e. there is some linear combination of this potential basis for each vector in the larger set.

The dimension is the cardinality of the set of basis vectors.

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