This has been a confusing topic, with half a dozen Wikipedia pages on the subject. Here I took some notes.

Tensors are sums of “products” of vectors. There are different kinds of vector products. The one used to build tensors is, naturally, the tensor product. In the Cartesian product of vector spaces \(V\times W\), the set elements are tuples like \((v,w)\) where \(v\in V, w\in W\). A tensor product \(v\otimes w\) is obtained by tupling the component bases rather than the component elements. If \(V\) has basis \(\{e_i\}_{i\in\{1,…,M\}}\) and \(W\) has basis \(\{f_j\}_{j\in\{1,…,N\}}\), then take \(\{(e_i,f_j)\}_{i\in\{1,…,M\},j\in\{1,…,N\}}\) as the basis of the tensor product space \(V\otimes W\). Then define the tensor product \(v\otimes w\) as

(1) \(\sum_{i,j} v_i w_j (e_i,f_j) \in V\otimes W\),

if \(v=\sum_i v_i e_i\) and \(w=\sum_j w_j f_j\). The entire tensor product space \(V\otimes W\) is defined as sums of these tensor products

(2) \(\{\sum_k v_k\otimes w_k | v_k\in V, w_k\in W\}\).

So tensors in a given basis can be represented as multidimensional arrays.

\(V\otimes W\) is also a vector space, with \(MN\) basis dimensions (c.f. \(V\times W\) with \(M+N\) basis dimensions). But additionally, it has internal multilinear structure due to the fact that it is made of component vector spaces, namely:

\((v_1+v_2)\otimes w = v_1\otimes w + v_2\otimes w\)
\(v\otimes (w_1+w_2) = v\otimes w_1 + v\otimes w_2\)
\(\alpha (v\otimes w) = (\alpha v)\otimes w = v\otimes (\alpha w)\)
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The latest problem from fakalin. Took some wrong turns and hints to solve it…

Given a polynomial \(f: \mathbb{Z}\to \mathbb{Z}\) with positive integer coefficients, how many evaluations of \(f\) does it take to obtain the polynomial?

(An \(f: \mathbb{R}\to \mathbb{R}\) polynomial with real coefficients would take the number of degrees plus 1 to specify, which, if it held in this case, would render the answer unbounded. But the correct answer in this case is surprisingly small.)
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An elementary, nice lemma relating to the optimization of multivariable functions says that the smallest “big thing” is still bigger than the biggest “small thing”, in other words,

\(\min_x \max_y f(x,y) \ge \max_y \min_x f(x,y)\).
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Here is a problem recently described to me. Apparently there is a more elegant solution (which may give more insight), but I don’t see it yet.

The problem: \(X, Y, Z\) are independent random variables uniformly distributed over [0,1]. What is the distribution of \((XY)^Z\)?
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Here’s a problem posed to me by a friend:

Consider an equilateral triangle ABC with edge length 1. At each vertex is an object that is capable of movement at exactly speed 1. Beginning at time 0, each of the three objects moves toward its initial adjacent neighbor object, as in a game of pursuit. Of course, by symmetry, the objects will meet at the incenter of ABC. The question: how far will they have traveled?
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It is said that 8 perfect shuffles of a standard deck of 52 cards results in an unshuffled deck. Why is that? Is it true only for 52 cards? What kind of shuffle is implied by this statement? Yesterday, we (high school math buddies) made pretty good sense of it.
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This question was floating around: can a determinant-1 diagonal matrix be factored into a product of triangular matrices with 1-diagonals, and how?

At least for the 2-by-2 matrix, this is answered in the affirmative here (Section 8.3). Here is a recapitulation.
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