True or false: Every set of orthogonal vectors in $\mathbb{R}^{{3^n}}$ is linearly independent 
True or false: Every orthogonal set of vectors in $\mathbb{R}^{{3^n}}$ is linearly independent.

I think that the answer is “false”, since an orthogonal set can contain the zero vector. But I'm not sure because of being so specific when writing “$\mathbb{R}^{{3^n}}$”. Should I have any other consideration to solve this exercise?
 A: It depends on the definition of an “orthogonal” system.
More explicitly, on whether such a system is allowed to contain the zero vector.

*

*If an orthogonal system is not allowed to contain the zero vector, then your counter-example does not work, and the given statement is true. (Assuming that the question is talking about orthogonality with respect to the standard inner product on $ℝ^{3^n}$.)


*If an orthogonal system is allowed to contain the zero vector, then your solution is correct and the statement false.
The definition of an “orthogonal set” from Wolfram MathWorld does not exclude the zero vector, so according to this convention your solution is correct.

In the context of an exam question I’d assume this to be a trick question meant to punish students who apply
$$
  \text{orthogonal} \implies \text{linearly independent}
$$
without checking the neccesary assumption of excluding the zero vector.
I don’t see how the exponent $3^n$ is relevant to the problem;
I imagine that it’s only there to provide the students with some extra confusion, or even to distract them from the actual trap behind the statement (the zero vector).
