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I'm trying to show that for any linear transformation $L:\mathbb R^{n}\to \mathbb R^{m}$ I can write

$$ L(v)=\sum_{k=1}^{n} \lambda_{k} \langle v, o_{k}\rangle u_{k}, $$

for all $v\in\mathbb{R}^n$, where $\{o_{1},...,o_{n}\}$ and $\{u_{1},...,u_{m}\}$ form orthogonal bases. The lambdas are scalars. Essentially I'm trying to prove the existence of these two orthogonal bases. I feel like I've seen something similar to this before, but I can't quite put my finger on it. Any suggestions?

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Maybe this: starting from an arbitrary basis, we can form aon orthonormal basis: en.wikipedia.org/wiki/Gram-Schmidt_orthogonalization –  Berci Oct 27 '12 at 16:24
    
Is $(v,o_k)$ supposed to be the inner product? –  wj32 Oct 27 '12 at 21:54
    
@wj32 Yes, it is the inner product. –  Alex Oct 27 '12 at 22:51
    
You can use \langle and \rangle for inner product, like $\langle u, v \rangle$. –  Ivo Terek Mar 9 at 21:00
    
This statement is the existence of singular value decomposition. If you need a hint, that is what you should google. –  Aaron Mar 9 at 21:16

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