What is the geometric interpretation of a multilinear subspace? [closed]

If we have two linearly independent vectors in n-dimensions, we know that they span a plane, for example. In general, they form a subspace.

I got introduced to multilinear algebra somehow, but I cannot picture in mind what kind of geometric construct a multilinear subspace is like? Is it possible for us to grasp it in such a manner or is it impossible for us to picture it in mind?

• What's the definition of "multilinear subspace"? I've never heard of such a thing... Commented Mar 1, 2019 at 12:43
• I did not made it up. "A multilinear subspace is defined through a multilinear projection that maps the input tensor data from one space to another (lower-dimensional space) " Direct sentence from Lu, Haiping, Konstantinos N. Plataniotis, and Anastasios N. Venetsanopoulos. "A survey of multilinear subspace learning for tensor data." Pattern Recognition 44, no. 7 (2011): 1540-1551. This sentence further cites this study: He, Xiaofei, Deng Cai, and Partha Niyogi. "Tensor subspace analysis." In Advances in neural information processing systems, pp. 499-506. 2006.
– user
Commented Mar 2, 2019 at 16:26
• If I knew what exactly a multilinear subspace corresponds to I would not be asking the question.
– user
Commented Mar 2, 2019 at 16:36

In multilinear algebra we impose condition of our function on vectors like $$T(v_1+v_1', v_2, v_3,\ldots, v_n)$$. This should be equal to $$T(v_1,v_2,\ldots, v_n) + T(v_1', v_2,\ldots, v_n)$$.