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[amp;quot;Ode to Joyamp;quot;, by Beethoven, plays to the end of the piano.] Traditionally, dot products are something thatamp;#39;s introduced really early on in a linear algebra course, typically right at the start. So it might seem strange that Iamp;#39;ve pushed them back this far in the series. I did this because thereamp;#39;s a standard way to introduce the topic, which requires nothing more than a basic understanding of vectors, but a fuller understanding of the role that dot products play in math can only really be found under the light of linear transformations. Before that, though, let me just briefly cover the standard way that dot products are introduced, which Iamp;#39;m assuming is at least partially review for a number of viewers. Numerically, if you have two vectors of the same dimension, two lists of numbers with the same lengths, taking their dot product means pairing up all of the coordinates, multiplying those pairs together, and adding the result. So the vec
