JOVANA
Library Glossary Getting Started Three Levels Fields How it works Mission
Join the mission
All guides

From Matrices to Maps

In Vol I a linear transformation was the thing a matrix did. We flip the emphasis: the map is primary, the matrix is a coordinate snapshot. We also gather all maps between two spaces into one new vector space.

The map was there all along

In Volume I you met the linear transformation T: V -> W as a rule that respects addition and scaling: T(au + bv) = aT(u) + bT(v). You then computed with its transformation matrix. The hidden lesson of this track is that those two objects are not the same. T lives on the spaces V and W and never mentions coordinates. The matrix only appears once you pick a basis for each side.

Why bother separating them? Because the same map gets infinitely many matrices, one per basis choice, and they all describe identical geometry. If we want to know what is really true about T — what it does, what it cannot do — we should phrase it without coordinates and let the matrix be a convenience, not the definition.

The matrix as a coordinate snapshot

Here is the recipe that turns a coordinate-free map into a matrix. Fix a basis B = (b1, ..., bn) of V and C = (c1, ..., cm) of W. To build the matrix of T relative to these bases, feed each basis vector of V through T and write the output in C-coordinates. Those coordinate columns, stacked side by side, are the matrix.

Map T: R^2 -> R^2,  T(x, y) = (x + 2y,  3y)
Standard basis B = C = ( (1,0), (0,1) )

Feed each basis vector through T:
  T(1,0) = (1, 0)   -> column [1; 0]
  T(0,1) = (2, 3)   -> column [2; 3]

Matrix relative to standard basis:
  A = [1, 2;
       0, 3]

Now switch V's basis to B' = ( (1,1), (1,-1) ):
  T(1, 1)  = (3, 3) = 3*(1,1) + 0*(1,-1)   -> [3; 0]
  T(1,-1)  = (-1,-3) = -2*(1,1) + 1*(1,-1) -> [-2; 1]

Same map T, NEW matrix (B' in, standard out):
  A' = [ 3, -2;
         0,  1]
One map, two matrices: changing the input basis changes the numbers but not the map.

Notice that the two matrices above are unmistakably different arrays of numbers, yet they encode exactly the same transformation of the plane. This is the whole motivation for studying the map directly: facts that survive every basis change are the structural ones, and the rest is bookkeeping.

All maps form a space

Now a leap. Take every linear map from V to W and put them in one bag. You can add two maps — (S + T)(v) = S(v) + T(v) — and scale a map — (aT)(v) = a*T(v). Both results are again linear. So the collection itself satisfies the vector space axioms: it is the space of linear maps, written L(V, W) or Hom(V, W).

Its dimension is delightfully simple: dim L(V, W) = (dim V)(dim W). The reason is the matrix recipe — once you fix bases, a map is exactly an m-by-n array of free numbers, and an m-by-n matrix space has dimension mn. So the abstract space of maps and the familiar space of matrices have the same size, basis by basis.