Let $\{v_1, v_2,...,v_n\}$ be a basis for an inner product space V. Show that the zero vector is the only vector in V that is orthogonal to all the basis vectors.
Let $v$ be any nonzero vector in V. $v$ is a linear combination of the basis vectors of V:
$$v = k_1v_1 + k_2v_2+ ... + k_nv_n$$
Therefore, for each basis vector $v_i$, the inner product with any vector in V is:
$$\langle k_1v_1 + k_2v_2 + \ldots + k_nv_n,v_i\rangle = \langle k_1v_1,v_i\rangle + \langle k_2v_2,v_i \rangle + \ldots + \langle k_iv_i,v_i \rangle + \ldots + \langle k_nv_n,v_i\rangle $$
This can only equal zero if all the basis vectors are zero. But this does not make sense because they would not form a basis of V and also because $v$ is non-zero.
Therefore no non-zero vector $v$ is orthogonal to all the basis vectors of V. However, for each basis vector $v_i$:
$$\langle 0,v_i \rangle = 0$$
Therefore the $0$ vector is orthogonal to all basis vectors $\{v_1, v_2,...,v_n\}$.
Verification and improvement would be appreciated.