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As the title states, if you have $A\subset V$ where $V$ is a vector space over an arbitrary field, does $A$ being linearly independent imply that the elements of $A$ are orthogonal?

Ali
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Complimentarity
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    Hmm, how do you define orthogonality in a general vector space over an arbitrary field? – vociferous_rutabaga Aug 18 '15 at 23:13
  • Given any linearly independent set of vectors on a finite dimension vector space, you can define a dot product on the space so that the vectors in the set are orthogonal. But in general, there isn't a single definition of orthogonal on a vector space - a vector space without a dot product does not have a notion of orthogonal. – Thomas Andrews Aug 18 '15 at 23:17
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    Even in $\mathbb{R}^n$ with the Euclidean inner product, this is not true; take $\mathbb{R}^2$ and $(1,0),(1,1)$. – Ian Aug 18 '15 at 23:24
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    The converse is not true either. In $\mathbb{F}_2^2$, $(1,1)$ and $(1,1)$ are orthogonal under the usual inner product, but they are linearly dependent. – Batominovski Aug 18 '15 at 23:32
  • @Ian In that case, would it be true to say that bases need not be orthogonal? If that is true, would it be correct to say that for an n-tuple space, there exists an n x n matrix of elementary row operations that can convert any basis into an orthogonal one? What about an n x m matrix space? – Complimentarity Aug 18 '15 at 23:34
  • Bases need not be orthogonal, that's correct. As for orthogonalization, look up Gram-Schmidt. – Ian Aug 18 '15 at 23:37
  • @Batominovski true, but the converse IS true if your inner-product is positive definite, like people usually require. – Jahan Claes Aug 19 '15 at 03:46
  • @Johan Claes. But how do you define positive-definiteness on an arbitrary field, as stated in the OP's problem statement? – Batominovski Aug 19 '15 at 03:48

1 Answers1

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It is not true. It is simple to find an example in $\mathbb{R}^2$ with the usual inner product: take $v=(1,0)$ and $u=(1,1)$, they are linearly independent but not orthogonal.

Kamila Andrade
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    Indeed, any two vectors in $\Bbb R^2$ that are not in the same (or opposite) direction, no matter how small the angle between them. – Lubin Aug 19 '15 at 01:21