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RE: Simple way to explain this to an undergrad?
Posted:
Jan 20, 2011 10:01 PM


I never associated linear combinations of matrices with matrix multiplication. Treating scalars as 1x1s, except in a few cases, generally advanced topics, just complicates matters. And it really has little to do with scalar * matrix anyway.
When I was in school (MS math '70), I did not think to question whether linear combinations of matrices was legitimate, because I saw it as analogous to linear combinations of vectors. And everyone easily understands THAT concept because it's very geometric. You need only consider the decomposition (a, b, c) = a e_1 + b e_2 + c e_3, where the e's are the unit directions, or "stretching" a vector by a positive factor, or reflecting it when multiplying by a negative factor.
Further, the scalar factors naturally out of the determinant (analogously to the way it factors from norm(scalar * vector)), as well as the inverse and transpose and matrix products, in precisely the way one would expect. The distribution laws work. Etc.
I've worked as a technical software developer for almost 40 years and have written several linear algebra packages along the way. I never hesitated to define a full set of scalarmatrix operators, often even including an operator "matrix / scalar"! Not one of the numerous engineers who used the packages, many with advanced degrees, ever questioned what they meant or whether they were legitimate.
All of those observations ought to give your students confidence that they aren't violating a Law Of Nature by doing what comes naturally.



