When a given function/equation is spanned by there respective space, i.e $\mathbb{R^n}$, $\mathbb{F^n}$, $\mathbb{T^n}$, etc. doesn't it mean that it forms a basis too?
For example:
The individual monomials $(sinx)^j(cosx)^k$ span $\mathbb{T^n}$, where $\mathbb{T^n}$ denotes the trigonometric polynomials, but it does not form a basis owing to identities stemming from the basic trigonometric formula $cos^2x + sin^2x = 1$.
Why is that? I thought if it spans it means there is a linear combination for that particular space, hence there will be a basis?