Why do mathematicians say that "let an operator be represented by a matrix" instead of operator is the matrix? 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"?
 A: 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.
A: The concept of matrix can be useful in other situation than for linear operators, and there exists linear operators that can not be represented by matrices.
The matrix is just an array of elements and are added and multiplied according to certain rules. For this it is only required that there is a meaningful definition for addition and multiplication for the elements. This means that the elements does not need to be real numbers, or even complex numbers. They can for example be integers (which may not seem that alarming, but it makes a difference). There's even more "odd" things one could put in a matrix as for example integers using modulo-n operations. These later cases means that the matrix do not represent a linear operator.
Then there are linear operators that does not operate on vector spaces of finite dimension or produce values from vector spaces of finite dimensions. For example a linear operator could map from or to a vector space of sequences or even functions. These can obviously not be represented as a matrix.
As for the normal case from linear algebra 101 one should remember that a linear operator is still the same no matter which basis for the vector space(s) you use, but the matrix representing the operator will be different depending on the choice of base. Even though you may think of a vector space as having a canonical base, this is not necessarily the case (it's better to think of them as being a space in it's own right and the base you select be arbitrary and no base is "better" than another).
A: The operator $T$ is a function from $\Bbb R^n$ to $\Bbb R^n$; an $n\times n$ matrix of real numbers is not a function. If one chooses the matrix $A$ properly, one can say that the function $T(x)=Ax$ for each $x\in\Bbb R^n$, but that is far from saying that the function $T$ is the matrix $A$.
Consider the more familiar case of real-valued functions on the reals. Specifically, consider the function $f(x)=ax$: this function is certainly not the same thing as the real number $a$. The relationship between $T$ and $A$ in the first paragraph is precisely similar to that between $f$ and $a$ in this paragraph.
