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Topic: RE: Simple way to explain this to an undergrad?
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Posts: 5
Registered: 11/13/10
RE: Simple way to explain this to an undergrad?
Posted: Jan 20, 2011 10:01 PM
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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

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 scalar-matrix 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.

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