If I have functions f(x) = sin(x), q(x) = x^2+1 and s(x) = x^2+sin(x) which are linear combinations, how do I represent these functions as vectors? The sin(x) part is throwing me off when constructing the augmented matrix.
These functions are linear combinations of a vector space V of continuous functions. I want to find a basis of V. I don't understand how to construct the matrix in the appropriate way to find the basis vector.
These functions each live in the vector space of functions of the form $p(x)+c\sin x$ for polynomial-or-vanishing $p$ and constant $c$. For $f$ we take $p=0,\,c=1$; for $q$ we take $p=x^2+1,\,c=0$; for $s$ we take $p=x^2,\,c=1$. In fact, $f,\,q,\,s$ live in a smaller space spanned by $1,\,x^2,\,\sin x$. You can write them relative to this basis, or any other basis, including of course $f,\,q,\,s$ themselves. So, choose whatever basis suits you.