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Image and kernel of a linear transformation. Online However, the true value of the dot product is realized when you relate this to the measurement of angles using trigonometry and the Law of Cosines.

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Composition of linear transformations and matrix multiplication. Students must fluently grasp the use of any of the following mathematical packages: Mathematica, or Maple, or Matlab. And you might be thinking, OK, Sal, fair enough. We relate transposes and dual spaces as follows. The notes are intended as review material, although many students like to read them as preparation before attending the lecture on the corresponding topics.

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