> A bit more precisely, in physics a tensor is an object that transforms a particular way during coordinate transformations
No offence, but that's a hideous definition :)
For me a (real) tensor is a function that takes an ordered set of N row vectors and M column vectors as arguments, and spits back a real number as a result. It has to be linear in its arguments. That's all folks!
By this token a matrix A is a tensor: it takes one row vector x, and one column vector y, and returns a real number xAy.
Similarly, a row vector x is a tensor: feed it a column vector y and you get the real number xy.
You can dress all this up in the language of linear functionals or n-forms, but at core that's what's going on.
Yes, that definition is fine for machine learning, but it's not quite complete for physics. To extend your definition for physics, a tensor is a function that takes an ordered set of N row vectors and M column vectors as arguments and spits back a real, coordinate-invariant number as a result.
I think you get coordinate invariance for free if you think of a vector as an object in its own right, rather than as a tuple in a coordinate system. But then I guess it's more accurate to speak of vectors and covectors than row vectors and column vectors.
No offence, but that's a hideous definition :)
For me a (real) tensor is a function that takes an ordered set of N row vectors and M column vectors as arguments, and spits back a real number as a result. It has to be linear in its arguments. That's all folks!
By this token a matrix A is a tensor: it takes one row vector x, and one column vector y, and returns a real number xAy.
Similarly, a row vector x is a tensor: feed it a column vector y and you get the real number xy.
You can dress all this up in the language of linear functionals or n-forms, but at core that's what's going on.