Most high school students in the United States learn about matrices and matrix multiplication, but they often are not taught *why* matrix multiplication works the way it does. Adding matrices is easy: you just add the corresponding entries. However, matrix multiplication does not work this way, and for someone who doesn’t understand the theory behind matrices, this way of multiplying matrices may seem extremely contrived and strange. To truly understand matrices, we view them as representations of part of a bigger picture. Matrices represent *functions* between spaces, called vector spaces, and not just any functions either, but **linear** functions. This is in fact why **linear algebra** focuses on matrices. The two fundamental facts about matrices is that *every matrix represents some linear function*, and *every linear function is represented by a matrix*. Therefore, there is in fact a one-to-one correspondence between matrices and linear functions. We’ll show that…

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