Learning new things in Linear Algebra

I always get excited when I learn something new. For example I recently learned that standard basis for polynomials is 1, x, x^2, x^3 ... and so on. For a matrix of size n x m there are n*m set of vectors which form the standard basis. This is something different from anything I learned in my first course in linear algebra. Here you don't have each vector as a tuple of some k things. So this leads to a fundamental shift in the way we think of functions. Now with just 3 basis 1, x and x^2 (note that either of these doesn't have components like for example basis e1,e2,e3 in R3 have e1 = (1,0,0), e2 - (0,1,0) and e3 = (0,0,1). Thus 1 is just 1, x is just x and x^2 is just x^2) one can think of a 3 dimensional space and each point of this space denotes a polynomial. For example the polynomial 3+7*x-21*x^2 is 3 units on axis 1, 7 units on axis x and -21 units on axis x^2.
The other fascinating concept is that of Linear Transformation. This tells you how you can map vectors from one vector space to another. Usually the notation for this is T:V->W. Where T denotes the linear transformation from vector space V to vector space W. Thus T takes input as vectors from the set V and maps it to the vectors in W. So T is just like a function however it doesn't take any vector it takes only those which satisfy the condition of T(x+y) = T(x) +T(y) and T(c.x)= c T(x) where x,y belongs to Vector over a field F and c belongs to F. At first shot one might not appreciate that use of the basis 1,x,x^2.. might enable one to do differentiation. Yes one can define a Transformation Matrix which can do operation like differentiation. I think it may be because when we think of differentiation we think that its not a linear transformation but its a basis which are non linear. What we are doing is just a c.x kind of operation. Other important concept which fascinated me was how we can map a matrix to a element or a vector with n component to a matrix. Lots of exciting things and that is all making me happy these days !! Yay yay for Linear Algebra !!