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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?

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    $\begingroup$ What's the definition of "multilinear subspace"? I've never heard of such a thing... $\endgroup$ Commented Mar 1, 2019 at 12:43
  • $\begingroup$ 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. $\endgroup$
    – user
    Commented Mar 2, 2019 at 16:26
  • $\begingroup$ If I knew what exactly a multilinear subspace corresponds to I would not be asking the question. $\endgroup$
    – user
    Commented Mar 2, 2019 at 16:36

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In calculus there is one variable calculus versus multi-variable calculus. In linear algebra the analogue is: study of function on ONE vector space (they should be linear transformations) versus function defined on many variables each coming from different vector space (I am deliberately silent about co-domain of these functions, could be any vector space in both the cases).

Now to carry the analogy further in multi-variate calculus we have the concept of partial derivatives. This means only one variable undergoes infinitesimal changes, and the others are fixed.

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 $T(v_1,v_2,\ldots, v_n) + T(v_1', v_2,\ldots, v_n)$.

This analogy with partial derivative should help you understand further.

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