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Let $B=\{e_1=(1,0,0), e_2=(0,1,0),e_3=(0,0,1)\}$ be a base of $\mathbb{R}^3$.

Does there exist an inner product $\langle\cdot,\cdot\rangle:\mathbb{R}^3\to\mathbb{R}$ such that $\langle e_1,e_1\rangle=2$, $\langle e_2,e_2\rangle=3$, $\langle e_3,e_3\rangle=4$, $\langle e_1,e_2\rangle=0$ and $\langle e_2,e_3\rangle=\langle e_1,e_3\rangle=1$?

This problem was taken of an Analisys book (first chapter). I think I need to know some property of inner product to solve it, but the author just defines inner product in the text. So, I would like know if there is a way to ensure the existence of an inner product in this kind of problem.

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  • $\begingroup$ This problem was taken of an Analisys book (first chapter). I think I need to know some property of inner product to solve it, but the author just defines inner product in the text. So, I would like know if there is a way to ensure the existence of an inner product in this kind of problem. $\endgroup$ – Pedro Jul 21 '13 at 18:38
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Hint: $$\begin{pmatrix} a_1 & a_2 & a_3\end{pmatrix} \begin{pmatrix} 2 & 0 & 1\\ 0 & 3 &1\\ 1 & 1 & 4\\ \end{pmatrix}\begin{pmatrix} b_1 \\ b_2 \\ b_3\end{pmatrix}$$

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  • $\begingroup$ As a minor issue, one still needs to prove that this matrix is positive definite, but this is trivial. $\endgroup$ – TZakrevskiy Jul 21 '13 at 18:31
  • $\begingroup$ Why does the $1\times 3$ look so strange? $\endgroup$ – Pedro Tamaroff Jul 22 '13 at 0:47
  • $\begingroup$ @Peter: That's simply how it's supposed to look; the columns are the correct width apart. The code $$\begin{pmatrix} a_1 & a_2 & a_3\end{pmatrix}\\ \begin{pmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3\\ \end{pmatrix}$$ produces $$\begin{pmatrix} a_1 & a_2 & a_3\end{pmatrix}\\ \begin{pmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3\\ \end{pmatrix}$$ $\endgroup$ – Zev Chonoles Jul 22 '13 at 0:56
  • $\begingroup$ The $a_i$ should be alittle higher, IMO. $\endgroup$ – Pedro Tamaroff Jul 22 '13 at 1:06
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Hint: Try to find an invertible matrix A whose columns $v_1,v_2,v_3$ satisfy the given conditions, and then define $<u, v>=(Au)\cdot (Av)$. For example, you could take $v_1=\begin{pmatrix}1\\1\\0\end{pmatrix}$ and $v_2=\begin{pmatrix}1\\-1\\-1\end{pmatrix}$, and then solve for $v_3$.

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