Prove or disprove the following statement about vectors in :

If for every vector , then .

*Proof.* Let . If for every we must, in particular, have . But if and only if . Hence, we must have

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Stumbling Robot

A Fraction of a Dot
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Prove that the only vectors whose dot product with all vectors is zero is the zero vector

Prove or disprove the following statement about vectors in :

If for every vector , then .

*Proof.* Let . If for every we must, in particular, have . But if and only if . Hence, we must have