The matrix of a linear map

Here is the result that ties the two halves of the course together: in finite dimension, every linear map is a matrix, and every matrix is a linear map.

A basis determines everything

Let f be linear and (e1, …, en) a basis of the source space. Every vector is written v = x1e1 + … + xnen, so by linearity:

f(v) = x1·f(e1) + x2·f(e2) + ... + xn·f(en)

In other words: if we know the images of the basis vectors, we know f everywhere. A linear map, though defined on infinitely many vectors, fits in n pieces of data.

The matrix: images as columns

We store those images as columns, in order:

              f(e1)  f(e2)  f(e3)
              |      |      |
        A =  [ .      .      . ]
             [ .      .      . ]

  column j of A = coordinates of f(ej)

And then, for every vector v with coordinates X:

f(v)  corresponds to  A X

Computing the image of a vector becomes a plain matrix product.

The example to know: the quarter-turn rotation

In the plane, the 90° anticlockwise rotation sends:

e1 = (1 ; 0)  ->  (0 ; 1)              ^
e2 = (0 ; 1)  -> (-1 ; 0)              |  e2 -> (-1;0)
                                       |
        [ 0  -1 ]                  ----+----> e1 -> (0;1)
   A =  [ 1   0 ]                      |

Check it on v = (2 ; 3):

[ 0  -1 ] [ 2 ]   [ -3 ]
[ 1   0 ]×[ 3 ] = [  2 ]       (2 ; 3) does rotate to (-3 ; 2)

A few other useful geometric matrices:

scaling by k               projection onto x-axis     reflection in x-axis
   [ k  0 ]                   [ 1  0 ]                  [ 1   0 ]
   [ 0  k ]                   [ 0  0 ]                  [ 0  -1 ]

Composing means multiplying

Here at last is the justification for the odd definition of the matrix product:

matrix of (f then g)  =  (matrix of g) × (matrix of f)

Matrix multiplication is composition of maps. This explains two earlier facts at once:

• non-commutativity: rotating then projecting is not projecting then rotating;
• the reversed order: f is applied first but written on the right, as in the notation g ∘ f.

Likewise, A⁻¹ is the matrix of the inverse map, when it exists.

Change of basis, in one sentence

The matrix depends on the chosen basis: the same geometric object has several matrices. Choosing the basis well means getting the simplest possible matrix — ideally a diagonal one. That is the whole point of the course on diagonalisation.

Summary

• A linear map is entirely determined by the images of the basis vectors.
• Those images, stored as columns, form the matrix of the map.
• Computing an image reduces to the product A X.
• Quarter-turn rotation: [[0, -1], [1, 0]].
• Composing two maps means multiplying their matrices, in the reverse order of application.
• The matrix depends on the basis; changing basis changes the matrix, not the map.