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I need to determine whether the following function is tensor on $\Bbb R^4$ and express it in terms of elementary tensors. Can someone please help me with it? I do not know what elementary tensor means either. $$f(x,y)=3x_1y_2+5x_2 x_3$$

Thanks in advance!

Marion Crane
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  • Hmmm, $\color{Blue}{x_1},\color{Blue}{x_2},\color{Blue}{x_3},\color{Green}{y_2}$: sure your variables are right? I assume you're checking if this is a bilinear map $\Bbb R^2\times\Bbb R^2\to\Bbb R$ (in which case elementary tensors I believe correspond to products of linear functionals). – anon Feb 25 '15 at 01:22
  • I am sure the variables are right. That's whats there. And it is written that I need to determine if it is a tensor on $\Bbb R^4$. I am using Munkres's analysis. – Marion Crane Feb 25 '15 at 01:26
  • If this is in a textbook can you give the page #? – anon Feb 25 '15 at 02:11
  • Sure. page 226 problem 4 @anon – Marion Crane Feb 25 '15 at 02:37
  • Ah, it means $\Bbb R^4\times\Bbb R^4\to\Bbb R$. Okay. – anon Feb 25 '15 at 03:00

1 Answers1

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I assume that $ f: \mathbb{R}^{4} \times \mathbb{R}^{4} \to \mathbb{R} $, as mentioned by anon.

Now, a tensor can be defined as a multi-linear mapping. Hence, if $ f $ were a tensor, it would have to satisfy $$ \forall \mathbf{x},\mathbf{y} \in \mathbb{R}^{4}, ~ \forall \lambda \in \mathbb{R}: \quad f(\lambda \cdot \mathbf{x},\mathbf{y}) = \lambda \cdot f(\mathbf{x},\mathbf{y}) \quad \text{and} \quad f(\mathbf{x},\lambda \cdot \mathbf{y}) = \lambda \cdot f(\mathbf{x},\mathbf{y}). $$ However, by choosing $ \mathbf{x} = (0,1,1,0) $, $ \mathbf{y} = (0,0,0,0) $ and $ \lambda = 2 $, we see that \begin{align} f(\lambda \cdot \mathbf{x},\mathbf{y}) & = 3 (\lambda x_{1}) y_{2} + 5 (\lambda x_{2}) (\lambda x_{3}) \\ & = 20; \\ \lambda \cdot f(\mathbf{x},\mathbf{y}) & = \lambda (3 x_{1} y_{2} + 5 x_{2} x_{3}) \\ & = 10. \end{align} Therefore, $ f $ is not a tensor.

Berrick Caleb Fillmore
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  • Thanks for your answer. What does elementary tensor means tho? Can you talk about that a bit? – Marion Crane Feb 27 '15 at 06:20
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    @MarionCrane: Hi Marion. An elementary tensor is defined as a multi-linear mapping $ g: \mathbb{R}^{4} \times \mathbb{R}^{4} \to \mathbb{R} $ that satisfies $$ \forall \mathbf{x},\mathbf{y} \in \mathbb{R}^{4}: \quad g(\mathbf{x},\mathbf{y}) = \phi(\mathbf{x}) \cdot \psi(\mathbf{y}), $$ where $ \phi $ and $ \psi $ are both linear functionals on $ \mathbb{R}^{4} $. We then write $ g = \phi \otimes \psi $. As $ f $ has been shown to be not a tensor, it cannot be a sum of elementary tensors. – Berrick Caleb Fillmore Feb 28 '15 at 17:58