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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 multiplying matrices corresponds to composing the functions that they represent. Along the way, we’ll examine what matrices are good for and why linear algebra sprang up in the first place.

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