How are specific linear maps defined? I'm revising for exams and a question that often crops up is: given a linear map $\mathcal{T}:\;\mathbb{R}^n\to\mathbb{R}^m$, describe how to represent $\mathcal{T}$ as a matrix relative to bases $\mathfrak{B}_n,\mathfrak{B}_m$ of $\mathbb{R}^n,\mathbb{R}^m$.
What confuses me is how $\mathcal{T}$ can be defined in the first place without referring to bases. Is it implied that $\mathcal{T}$ is defined under the standard bases? Or at least under some bases? (if so then I know how to answer the question)
 A: It's hard to distinguish between vectors and their representations, as well as between linear maps and their representations if you only look at the vector spaces $\mathbb{R}^n$ and $\mathbb{C}^n$, because they look the same there.
But the idea becomes clear if you pick a vector space where the actual vectors aren't $n$-tuples, but instead something else. A good example is the vector $P_n$ space of real-valued polynomial functions of degree at most $n$. This vector space contains polynomial functions of the form $$
  x \mapsto a_n x^n + \ldots a_1 x + a_0 \text{ for arbitrary } a_1,\ldots,a_0 \in \mathbb{R}
$$
Multiplication of vectors with scalars and sums of vectors are defined point-wise, i.e. $$
  (p + q) = x \mapsto p(x) + q(x) \text{, } \lambda p = x \mapsto \lambda p(x) \text{.}
$$
Since every polynomial function of degree at most $n$ can be written at a sum of the $n+1$ monomial functions $x \mapsto$, $x \mapsto x$, $x \mapsto x^2$, ... , $x \to x^n$, the vector space obviously has dimension $n+1$.
It's easy to define a linear map on this vector space without any references to a matrix. Let $D$ be the (linear!) map that maps a polynomial function to it's derivative, i.e. $$
  D \,:\, P_n \to P_n \,:\, p \mapsto p'
$$
You can similarly define linear functionals, i.e. linear maps from $P_n$ to $\mathbb{R}$ without references to a basis. For any real number $\mathbb{c}$, you can define the map $$
  e_c := P_n \to \mathbb{R} \,:\, p \mapsto p(c)
$$
A good exercise for you would be to 


*

*Show that $P_n$ is indeed a vector space

*Show that $D$ is a linear map $P_n \to P_n$. Is that map injective? surjective? Find $\ker D$ and $\textrm{img } D$.

*Find a basis of $P_n$. (The text above actually already states one)

*Find a matrix representation of $D$ in that basis


If you have already covered dual spaces, you could additionally


*

*Find the dual space $P_n'$. Find a basis in terms of suitable chosen $e_c$.

*Find the dual basis of the basis of $P_n$ you found earlier.

*Find a matrix representation of $D'$, the dual of $D$, in that basis 

A: This might seem off-topic at first, but if you read on it will be relevant. . . I hope ;-)
Let's start with a true-or-false question: with respect to the standard basis, any vector is the same as its coordinate vector.  For example,
$$\hbox{the coordinate vector of}\quad\pmatrix{2\cr3\cr}\quad\hbox{is}\quad
  \pmatrix{2\cr3\cr}\ .$$
True or false?  I would say the statement is false.  Sure, the numbers involved are the same, but I would not regard the vectors as being really the same, because they are conceptually different.  You can think of the actual vector as a physical object; for example, you can conceptualise it as an arrow.  However, the coordinate vector should not be thought of as a physical object: you should rather think of it as a recipe, or a set of instructions, which says "take $2$ times the first basis vector plus $3$ times the second".
So, a definition like $T({\bf v})=A{\bf v}$ need not be taken to refer to any particular basis at all: you can regard it as saying, "take the physical vector $\bf v$ and perform on it the operations indicated by $A$".  To take a specific example, $A$ might be a rotation matrix: then what we have to do to any vector is to rotate it through a certain angle about the axis specified by another vector, and this can be done without needing to call on any particular basis.
Hope this helps!
A: Well , the linear map $\mathcal{T}:\;\mathbb{R}^n\to\mathbb{R}^m$ is a function for which the following two conditions are satisied:
(1) $\mathcal{T}(x+y) = \mathcal{T}(x) + \mathcal{T}(y)$ for $x,y \in \mathbb{R}^n$
(2) $ \mathcal{T}(\alpha x)= \alpha \mathcal{T}(x)$ for $\alpha \in \mathbb{R}$
We can represent this linear map by a $m\times n$ matrix. We can represent, say, an element $v$ of $\mathbb{R}^n$ using a column matrix of size $n$. When we multiply $v$ by an $m \times n$ matrix, the output is a column matrix of size $m$, which can be identified with some element of $\mathbb{R}^m$.
Our task is to find such a matrix $A$, which has the same effect on the elements of $\mathbb{R}^n$ (identified as column matrices), as does the linear map $\mathcal{T}$. As to how to go about constructing such a matrix, there are several sources availaible online, like the khan academy, MIT opencourse, etc.
