In the book I am referring (Jim Hefferon) Exchange Lemma for the basis is given as:
Assume that $B =\big< \overrightarrow{\beta_1}, \overrightarrow{\beta_2},...\overrightarrow{\beta_n}\big> $ is a basis for a vector space, and that for the vector $\overrightarrow{v}$ the relationship $\overrightarrow {v}$ = $c_1 \overrightarrow{\beta_1}+c_2\overrightarrow{\beta_2}+...+c_n\overrightarrow{\beta_n}$ has $c_i \neq 0$. Then exchanging $\overrightarrow{\beta_i}$ for $\overrightarrow{v}$ yields another basis for the space.
Intuitively speaking, we are replacing one of the elements of the basis (which is linearly independent by definition) with an element that is linear combination of other elements, so now the basis does not remain linearly independent and so is no more a basis, but it is not happening as per my argument, so what am I missing?