TITLE: Why do mathematicians say that "let an operator be represented by a matrix" instead of operator is the matrix?
QUESTION [8 upvotes]: For example, look at this sentence from Perko's text on dynamical system
"It follows from Cauchy Schwarz inequality that if $T \in L(R^n)$ is represented by the matrix $A$ with respect to the standard basis for $R^n$ $_\cdots$" pg 11
What does it mean for a $T: R^n \to R^n$ to be represented by a matrix? Isn't it by definition that $T: R^n \to R^n$ is equivalent to an n by n matrix? Can someone translate exactly what it means by "represented" versus "not represented"?

REPLY [9 votes]: The number after nine is represented in the usual base system by the sequence of digits $10$, but it isn't a sequence of digits. Properties like "two digits long" are properties of the representation, while properties like "has four factors" are properties of the number. We could represent the number in binary, as $1010$, and that would be just as valid.
Similarly, a linear transformation $T$ may be represented by a matrix $M$, but it isn't the matrix. Properties like "upper triangular" are properties of the representation, while properties like "injective" are properties of the linear transformation. In a different basis, $T$ might be represented by a matrix other than $M$, and that representation would be just as valid.