I want to find $m,l,n$ and $k$ and $(A_1,B_1),(A_2,B_2)\in V\times W$ such that $A_1B_1+A_2B_2\neq A_3B_3$ for any $(A_3,B_3)\in V\times W$.

Question

I am reading "Tensor Algebra" by Takeo Yokonuma (in Japanese).

Problem 2 (on p. 329)
Let $V,W$ and $U$ be vector spaces over $k$, where $k$ is a field.
Let $\mathcal{L}(V,W;U)$ be the set of all bilinear mappings from $V\times W$ to $U$.
Let $\Phi\in\mathcal{L}(V,W;U)$.
Let $S:=\{\Phi(v,w)\mid v\in V, w\in W\}$.
Prove that there exist $V,W,U$ and $\Phi$ such that $S$ is not a subspace of $U$.

My attempt is here:

Let $M(m,n;k)$ be the set of all $m\times n$ matrices whose elements are elements of $k$.
Let $V:=M(m,l;k),W:=M(l,n;k)$ and $U:=M(m,n;k)$.
Then, $\Phi:V\times W\ni(A,B)\mapsto AB\in U$ is a bilinear mapping.
I want to find $m,l,n$ and $k$ and $(A_1,B_1),(A_2,B_2)\in V\times W$ such that $A_1B_1+A_2B_2\neq A_3B_3$ for any $(A_3,B_3)\in V\times W$.
I know the following propositions. (I don't know that the following propositions help me or not.)

Let $A,B\in M(m,n;k)$.
Then, $\operatorname{rank}(A+B)\leq\operatorname{rank}A+\operatorname{rank}B$.

Let $A\in M(m,l;k), B\in M(l,n;k)$.
Then, $\operatorname{rank}A+\operatorname{rank}B-l\leq\operatorname{rank}(AB)\leq\min\{\operatorname{rank}A,\operatorname{rank}B\}.$

Answer 1

Another solution that also uses spaces of matrices, but which is more simple and valid for any field $k$.

It is based on the equality:

$$\begin{pmatrix} 1\\ 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \end{pmatrix} + \begin{pmatrix} 0\\ 1 \end{pmatrix}\begin{pmatrix} 0 & 1 \end{pmatrix}= \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}+\begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}$$ The last matrix, i.e. the identity, can't be written as a product $AB$ where $(A,B) \in k^{2 \times 1} \times k^{1 \times 2}$.

Answer 2

Prove that an augmented matrix $\begin{matrix} (A_1&A_2)\end{matrix} \begin{pmatrix} B_1\\B_2 \end{pmatrix} = A_1B_1+A_2B_2$

The above question helped me.
Thank you very much.

I found the following example:

$\operatorname{rank}(A_3B_3)\leq\min\{\operatorname{rank}A_3,\operatorname{rank}B_3\}\leq 2$ for any $3\times 2$ matrix $A_3$ and any $2\times 3$ matrix $B_3$.