Ever since the dawn of the computer age, researchers have been trying to find an optimal way of multiplying matrices, a fundamental operation that is a bottleneck for many important algorithms. Faster matrix multiplication would give more efficient algorithms for many standard linear algebra problems, such as inverting matrices, solving systems of linear equations, and finding determinants. Eve...

Recall that in a commutative ring 〈R, 0, 1, +,×〉, • 〈R, 0,+〉 is an abelian group, • 〈R− {0}, 1,×〉 is an abelian semigroup, and • multiplication distributes over addition. The same structure is a field if, in the second clause, 〈R − {0}, 1,×〉, is a group; in other words, every nonzero element has an inverse. A vector space V over a field F is an abelian group 〈V, 0, +〉 with an operation of “scal...

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