Linear algebra with vectors plz help?

find the dimension of vectorial subspace C(-π,π) that been composed by the vectors: sinx*cosx, sin2x, cos2x, sin^2(x), cos^2(x). please explain first what is ''dimension of vectorial subspace'' and please analyze the whole process

Linear algebra with vectors plz help?
In linear algebra, "dimension" means no. of linear indepent vectors in the subspace.





definition of linear indepent vectors: Let c1,c2....cn be any constant. If vecotrs v1,v2....vn are linear independent to each other, then c1v1+c2v2+...+cnvn=0


iff c1=c2=...=cn=0 (In other words, v1,v2,...,vn are linear independent to each other if each of v1,v2,...,vn cannot be expressed as linear combination of other vectors)





***In this question, you have to recognize which of these vectors are independent





eg.


sinxcosx and cos 2x=0 are linear independent to each other, but sinxcosx and sin2x are not since sin2x=2cosxsinx





In order to get a better observation to no. of linear independent vectors, let's expand sin2x and cos2x by trig. formula:





sin2x=2sinxcosx


cos2x=cos^2(x)-sin^2(x)


Obviously, sin2x and cos2x are not linear independeent vectors since sin2x and cos2x can be expressed as linear combination of sinxcosx and (cos^2(x),sin^2(x)) respectively.





Is their relationship between sinxcosx, cos^2(x) and sin^2(x)? No. Hence, these three vectors are linear independent to each other.





Since there are 3 linear independent vectors, hence dimension=3

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