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Edit: Exact Question. my question is b part enter image description here$\phi:E\times F\to G$ be bilinear
$\psi:E\times F\to H$ be bilinear
Given $N_1(\phi)\subset N_1(\psi)$ and $N_2(\phi)\subset N_2(\psi)$ Show that there exist a linear function $f:G\to H$ such that $f\cdot\phi=\psi$ where $N_1(\phi) = \{x|\phi(x,y)=0\forall y\in F\}$ and similarly $N_2$ for the second coordinate

My "counterexample"
$F=\mathbb R^3$
$H=E=\mathbb R^2$
$G=\mathbb R^5$
$\phi(e,f)=(e_1f_1,e_1f_2,e_1f_3,e_2f_1,e_2f_2)$
$\psi(e,f)=(e_2f_2,e_2f_3)$
$N_1(\phi)=\{(0,0)\}\subset\{(x,0)\}=N_1(\psi)$
$N_2(\phi)=\{(0,0,0)\}\subset\{(x,0,0)\}=N_2(\psi)$

Please point out the mistake in the "counterexample". Also please provide hints for the problem

Anvit
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  • It seems as if $G$ and $H$ should be subsets of $\mathbb R$ (or some other field). Otherwise what does $f\cdot g$ mean? $f$ and $\psi$ both take value in $H$. – MPW May 24 '19 at 17:53
  • @mpw I'm not sure what $f\cdot g$ means, i don't have it anywhere – Anvit May 24 '19 at 18:04
  • Huh? Then how can you have a counterexample? A counterexample would be a scenario satisfying the hypotheses but for which no such linear function $f:G\to H$ exists. – MPW May 24 '19 at 18:05
  • @mpw I'm saying no linear function would exist $G\to H$ – Anvit May 24 '19 at 18:10
  • @MPW Why do $G$ and $H$ need to be subsets of a field? $f\cdot g$ likely means composition of functions. – Michael Burr May 24 '19 at 18:13
  • I know what $f\cdot g$ in general is but I do not have a $g$ anywhere in the question – Anvit May 24 '19 at 18:14
  • I should have said $f\cdot \phi$. But composition doesn't make sense in this case, given what the spaces are – MPW May 24 '19 at 18:20
  • why not? phi is to $G$ and f is from $G$. what is the problem? – Anvit May 24 '19 at 18:49
  • Okay, I think you're right, my bad – MPW May 24 '19 at 20:20

1 Answers1

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Your counterexample is correct, it shows the claim made before cannot be correct (there is no way to get $e_2f_3$ as a linear combination of the components of $\phi(e,f)$). Are you sure you copied the problem correctly, especially the definition of $N_1, N_2$?

Ingix
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