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!
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.