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Explain yourselves, comp sci.
(mander.xyz)
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this post was submitted on 28 Apr 2024
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Wouldn't N by M be a tensor? Magnitude and direction only need one entry per DOF.
Every vector is a tensor. Matrices are vectors because m by n matrices form vector spaces. Magnitude and direction have nothing to do with the definition of vectors which are just elements of vector spaces.
All vectors are tensors but not vice versa. And every page/definition of vector I've seen references magnitude and direction, even the vector space page you linked.
It looks like "vector" commonly refers to geometric vectors which is what most folks in this thread are discussing.
Would N by M vectors be imaginary, where each DOF has real and imaginary components?
Continuous functions on [0,1] are vectors. Magnitude and direction are meaningless in that vector space, usually called C[0,1]. Magnitude and direction are not fundamental properties of vectors.
n by m matrices (and the vector spaces to which they belong) are perhaps best thought of similarly to functions and function spaces. Not as geometric objects, but as linear transformations (which they are).