Wednesday, October 18, 2006

"Live 'n' learn," or, "D'oh!"

One of the drawbacks of running a class in this format is that it's often hard to tell just how much we can "push the envelope."

We've now spent a bit of time talking about linear transformations, and I daresay most of the folks in the class are pretty adept at (1) testing algebraically whether or not a given function is a linear transformation using the defining characteristics of linear transformations, (2) determining the action of a linear transformation algebraically, given its action on a set of basis vectors, and (3) setting up a matrix which performs the given linear transformation. I know some of us have even begun to tackle the problems from Section 3.4, wherein we consider linear transformations in arbitrary "exotic" vector spaces.

But there's a word which appears frequently in the previous paragraph, and that word is "algebraically": indeed, so far as a class (some of the MATH 365 folks, like the crystal-gazers and the computer graphics programmers, whose research topics are quite geometric in nature, are excepted here) we have not considered linear transformations from a geometric point of view.

While I've already received one electronic request to remedy this oversight on Friday when we have a linear transformation free-for-all, it will do no good as far as today's quiz was concerned, in which I asked a rather bold question requiring the student to translate a geometric action into a linear transformation, and thence into a matrix.

While it was an ambitious question to ask, I'm glad to see that (a) a fair percentage of people in the class answered the first portion of the question nearly (or entirely!) correctly, and (b) an even more sizable chunk of the class mastered the second portion of the question splendidly, viz., constructing a matrix to mimic the transformation once its action on a basis was known.

In retrospect, I feel that the quiz was unfairly difficult (my bad), but I hope that all will soon come to see that the question is not an unreasonable one.

Onward! let us march, into a maelstrom of matrices and a hail of linear fire, as Friday brings us to consider linear transformations as they relate to every other aspect of a vector space's structure. Onward, onward, ONWARD!

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