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In the area of bilinear forms, my lecture notes say that there is a basis $\{e_i\}$ of $V$ with respect to which $\tau (e_i, e_j) = \delta_{ij}$ where $$\delta_{ij} = \begin{cases} 1 & i=j \\ 0 & i \neq j \end{cases}$$ and that a basis of a Euclidean space $V$ with this property is called an orthonormal basis of $V$. Does this mean that there is only one 'orthonormal basis' which is $(1,0,\dots,0), (0,1,\dots,0),\dots$ etc?

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    No. For example, $\{(1/\sqrt2, 1/\sqrt 2)(1/\sqrt2, -1/\sqrt 2)\}$ is an orthonormal basis of $\Bbb R^2$.2012-04-06
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    Note the condition $\tau(e_i,e_j)=\delta_{ij}$ describes how the basis elements act on each other; it is not describing what the basis elements look like exactly.2012-04-06
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    http://en.wikipedia.org/wiki/Orthonormal_basis2012-04-06

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