Prove that a Vector Orthogonal to an Orthonormal Basis is the Zero Vector.

Question

Stuck in this proof.

Let W be an inner product space (with unspecified inner product, $\langle\vec x, \vec y\rangle$), and with orthonormal basis $B = \{\vec w_1, \vec w_2, \ldots ,\vec w_n\}$.

Suppose that $\vec x$ is orthogonal to $\vec w_i$ for each 1 $\leq i \leq n$.

Prove that $\vec x$ is the zero vector.

Any help is appreciated.

Answer 1

Since $\{\mathbf{w_i}\}$ is a basis, any vector can be write as: $$ \mathbf v=\sum_i x_i \mathbf{w_i} $$ now, from orthogonality, we have: $$ \mathbf v \cdot \mathbf {w_i}=0 \quad \forall i \iff x_i=0 \quad \forall i $$