Show that $(2, 0)$ tensor is not a product of two vectors The problem I am trying to solve is:

Show that a general $(2, 0)$ tensor $K$, in $n$ dimensions, cannot be written as a direct product of two vectors, $A$ and $B$, but can be expressed as a sum of many direct products.

I was thinking that $(2, 0)$ tensor is a product of two vectors, but apparently not. Could someone give some intuition into this?
Thank you!
 A: Consider the vector space $\mathbb{R}^2$ with the standard basis $\{ e_1, e_2 \}$. Then tensoring any two elements of $\mathbb{R}^2$ together gives:
$ (a_1e_1 + a_2e_2) \otimes (b_1e_1 + b_2e_2) = \sum_{i, j} a_ib_j(e_i \otimes e_j) $
Can we tensor two elements of $\mathbb{R}^2$ together to get $e_1 \otimes e_1 + e_2 \otimes e_2$? Matching coefficients in the above sum we see that we would have to have $a_1b_1 = a_2b_2 = 1$. But then $a_1b_2 \neq 0$, and so we would have an $e_1 \otimes e_2$ term coming out as well. In other words, this is an impossible task!
Tensors that can be formed from products of elements in $\mathbb{R}^2$ are called the elementary tensors of $\mathbb{R}^2 \otimes \mathbb{R}^2$. Your question is essentially "Are all tensors elementary?", and as this example shows the answer is a definite no. The rest can only be realised as a sum of elementary tensors.
As for intuition, I find it helpful to think of the following situation. Consider the ring of all polynomials in $X$ and $Y$ with real coefficients, denoted $\mathbb{R}[X, Y]$. Many of these can be described as the product of a polynomial in $X$ and a polynomial in $Y$. For instance, $XY + X + Y + 1 = (X + 1)(Y + 1)$. On the other hand, $X + Y$ cannot be represented as a product like this.
This is not too surprising, because we are actually working with a tensor product here too: $\mathbb{R}[X, Y] = \mathbb{R}[X] \otimes \mathbb{R}[Y]$.
