Can someone please explain the difference between a matrix and linear transformation. I know that they can be the same thing, but have been told that they aren't the same thing necessarily. So my question is what is the difference?
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On it's own a matrix is just an array of numbers. But from a matrix you can get a linear transformation, just by left multiplying. Similarily, on it's own a linear transformation is just a map. But every linear transformation has a matrix representation. The associated linear transformation of this matrix (left multiplication) is the original transformation. So on their own, just as entities, a matrix is an array of numbers and a linear transformation is a map. But mathematically speaking they are isomorphic (i.e. the same thing). There is a correspondence between the two.
It's kind of like asking, what is the difference between the 3 people, Bob, Bill and Mary and the set $\{1,2,3\}$. On their own, one is a set of people, very different then a set of numbers. But there is a one to one correspondence between these 3 people and these 3 numbers.
JonHerman
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