Home page for accesible maths 6 Linear transformations

Style control - access keys in brackets

Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

6.4. Transformations of the Euclidean space, 3

So far we have mostly only been considering linear transformations of 2, even though the definitions are stated for n. In this section we will briefly consider linear transformations of 3, and in particular we would like to understand how to think about them as 3×3 matrices.

As usual, we will consider the standard basis vectors

e1=(100),e2=(010)ande3=(001)in 3.

Analogous to the 2-dimensional case, we define the scalars aij for 1i,j3 by the equations

T(e1) =a11e1+a21e2+a31e3
T(e2) =a12e1+a22e2+a32e3
T(e3) =a13e1+a23e2+a33e3

and write A=(aij). So A is the matrix associated to the linear transformation T.

Conversely, every matrix AM3() defines a linear transformation T:33 by T(v):=Av, via matrix multiplication. In this way, we have a bijective correspondence between linear transformations and matrices, as described in Section 6.2.

Example 6.4.1.

    1. (a)

      Any translation

      T:(xyz)(x+ay+bz+c)with(abc)(000)

      is not a linear transformation.

    2. (b)

      Let T:22 be a linear transformation of the plane and A=(aij) its associated 2×2 matrix. Then the map

      S:(xyz)(a11x+a12ya21x+a22yz)=(A(xy)z)

      is a linear transformation. Its associated matrix is

      B=(a11a120a21a220001)=(A00001).

      In particular, we can take T=Rθ or T=Hθ as in Proposition 6.2.8 and 6.2.9. They are given by the matrices

      Rθ=(cosθ-sinθ0sinθcosθ0001)andHθ=(cos2θsin2θ0sin2θ-cos2θ0001).
    3. (c)

      The reflection in any plane is a linear transformation. For instance, the reflection in the xy-plane, which fixes e1,e2 and sends e3 to -e3 is given by the diagonal matrix (10001000-1).