Showing that a function is a tensor

Question

I am trying to solve question 4 in Munkres Analysis on Manifolds section 26.

The question is determine if the following is a tensor on $\mathbb{R}^4$ and express those that are in terms of the elementary tensors on $\mathbb{R}^4$.

$f(x,y) = 3 x_1 y_2 + 5 x_2 x_3$

I am trying to follow the definition of a tensor (which I don't think I fully understand) to show this. I know I must find a function $T: \mathbb{R}^4 \rightarrow \mathbb{R}$ that is linear on the $i^{th}$ variable. Since there are only 2, then I believe I have to somehow write $f$ as a projection. I'm just not sure where to go from here.

Any hints would be great, thanks!

Answer 1

Perhaps I am missing something, but you have $f(\lambda x,0) = \lambda^2 f(x,0)$. Hence it is not linear in $x$, hence not a tensor.

If $f$ was a tensor, it would have the form $f:\mathbb{R}^4 \times \mathbb{R}^4 \to \mathbb{R}$, and be linear in each variable separately, that is the functions $w_1 \mapsto f(w_1,y)$ and $w_2 \mapsto f(x,w_2)$ would be linear for all $x,y$.

Answer 2

edit: Oops, I misread it as $ f(x, y) = 3x_1y_2 + 5x_2y_3$ . But still, the answer serves as an example of how to verify if it's a tensor.

Just compute $ f(x+z, y) = 3(x_1+z_1)y_2 + 5(x_2+z_2)y_3 = 3x_1y_2 + 3z_1y_2 + 5x_2y_3 + 5z_2y_3 = f(x, y) + f(z, y) $
Repeat for $ f(x, y+z) $, $f(cx, y) $ and $f(x, cy) $to verify that it's a tensor.