Definition 1: The vectors $v_1, v_2, . . . , v_n$ are said to span $V$ if every element $w \in V$ can be expressed as a linear combination of the $v_i$.
Let $v_1, v_2, . . . , v_n$ and $w$ be vectors in some space $V$. We say that $w$ is a linear combination of $v_1, v_2, . . . , v_n\;$ if $\;w = \lambda_1v_1 +\lambda_2v_ 2+ · · · +\lambda_nv_n$ $\;$for some scalars $\lambda_1,\lambda_2, . . . ,\lambda_n$.
Definition 2: We say that vectors $v_1, v_2, . . . , v_r$ are linearly independent if the only solution of $\lambda_1v_1 +\lambda_2v_2 + · · · +\lambda_rv_r = 0\;$ is given by$\; \lambda_1 = \lambda_2 = · · · = \lambda_r = 0\;$.
Definition 3: If $v_1, v_2, . . . , v_n$ are linearly independent and span $V$ we say that they form a basis of $V$ . The number $n$ is called the dimension.
My question: Does the Definition 3 of basis tell us that $\lambda_i$ has to be zero? Then $w=0$