If I remember my Algebra class correctly, this is in general not true because, for example, the addition isn't closed. So if I take:
$B:\mathbb{R}^2 \times \mathbb{R}^2\to \mathbb{R}^{2^2}\cong \mathbb{R}^2 \otimes \mathbb{R}^2$
$(m,n)\mapsto mn^t$
The Image of B isn't closed under Addition since the resulting Matrix of B(m,n) has either Rank 1 or 2.
Is this correct, or am I making a mistake along the way?