Let denote the unit coordinate vectors , , and respectively in . Prove that are linearly dependent, but any three of them are linearly independent.

*Proof.* First, any four vectors in are linearly dependent by Theorem 12.10 (page 466 of Apostol). Then we show that any set of three of these are independent.

First, are independent since

Then, are independent since if

then

The independence of the other two sets of three vectors follows identically