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The claim is James Sylvester and Arthur Cayley are the fathers of Linear Algebra. I can find the various parts that Cayley contributed to Linear Algebra, but there is not much on the contributions made by Sylvester. Does anyone have this information and direct me to resources on this? I only want his contribution to LA.

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Bill Dubuque's answer to this other question will also shed some light. I quote Bill:

Sylvester did similarly for many mathematical terms, i.e. coined them or used them in a "new or unusual ways" mathematically.

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As Chris points out in his comments, Sylvester coined the term matrix in 1850. According to the this, Sylvester also began the usage of annihilator, canonical form, Hessian, discriminant, Jacobian, minor and Nullity.

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This may be more than you want, but Karen Parshall has written two history of mathematics books on Sylvester which definitely include the information you want.

James Joseph Sylvester: Life and Work in Letters (Oxford University Press, 1998)

James Joseph Sylvester: Victorian, Jew, Mathematician (Johns Hopkins University Press, 2006)

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