Beyond Numeracy (Vintage) by Paulos John Allen

Author:Paulos, John Allen [Paulos, John Allen]
Language: eng
Format: epub
ISBN: 9780307833334
Publisher: Knopf Doubleday Publishing Group
Published: 2013-05-29T04:00:00+00:00

These transformations do not commute, and thus neither do the matrices representing them.

Performed in succession, these transformations of vectors needn’t be commutative. For example, a rotation of a vector followed by a reflection of the resultant vector isn’t always the same thing as a reflection followed by a rotation. (To see this, imagine a vector pointing northeast being rotated 45 degrees counterclockwise so that it’s pointing north. If this vector is now reflected about the east-west axis, it will end up pointing south. If instead the reflection of the vector about the east-west line occurred first, followed by a counterclockwise rotation of 45 degrees, the vector would end up pointing east.) Thus the matrices representing these rotations, reflections, and other transformations needn’t commute either; A*B doesn’t always equal B*A.

Matrices and vectors play a primary role in linear algebra as well as in many other areas of applied mathematics. The failure of commutativity partially explains why matrices are important in quantum mechanics, where the order in which two measurements are made affects the outcome. Tensors, a natural generalization of matrices, are a major ingredient in the mathematical formulation of general relativity theory. Perhaps the etymology of the word “matrix” is not so inappropriate after all.

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