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I'm not asking how matrix multiplication is done. I know how to multiply matrices and can look at two matrices and say whether or not they can be multiplied. I just want to know what matrix multiplication reveals to us.

Murph Jones
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    Do you know what a vector space is? Do you know what a linear transformation is? This question has rather different answers depending on whether these are familiar terms. – Milo Brandt Jun 28 '18 at 01:59
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    Matrices are representations of linear operators on a vector space, and matrix multiplication corresponds to composing two operators. By studying matrix multiplication, one can understand the effect of acting with multiple operators. For example, the fact that matrix multiplication does not always commute tells us that the order in which linear operators are applied to a vector will usually matter. –  Jun 28 '18 at 01:59
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    I started on page one of a linear algebra text book and I just got to multiplication. I haven't gotten to vector space yet but it's in the index and I'm motivated so I see myself getting to that page and beyond. Feel free to answer this question with any terminology and if I don't understand initially then I will later! – Murph Jones Jun 28 '18 at 02:02
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    As noted above, matrix multiplication is the composition of linear operators between vector spaces. This idea generalizes to the composition of linear operators on infinite dimensional vector spaces (the horror! the horror!). The modern model of quantum mechanics is, at its heart, the study of these kinds of operators. Hence (to give an out-there reason why one might care) understanding matrix multiplication is the first step to understanding a really important concept in modern physics and, perhaps, learning a bit about how the universe works. – Xander Henderson Jun 28 '18 at 02:09
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    The text begins matrix multiplication before it introduces vector spaces? No wonder you are confused. As others have said, the the matrix is a linear operator. It is a special kind of function. And matrix multiplication is the composition of these operators. – Doug M Jun 28 '18 at 02:21
  • You can also see the result of a matrix multiplication as the collection of all the inner products between rows and columns of two matrices. – Gonzalo Benavides Jun 28 '18 at 02:54

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This is a pretty broad question, matrix multiplication does many things and I'm sure I cannot give a complete answer as to all the reasons why its useful. Here is one pretty elementary reason why its useful. Recall if I have two functions say $f,g: \mathbb{R}^n \to \mathbb{R}^n$, I can compose them, making a new function $h = f \circ g$. Now if we suppose both $f,g$ are linear operators, and using bases, find a matrix form for them, then $f \circ g = f \cdot g$ as matrices. I encourage you to try to prove this to yourself so you can see its use

Sheel Stueber
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