Transformation Geometry — Math 331

March 12, 2004

The Affine Matrix of an Affine Transformation

Recall from the study of linear algebra that if f is a linear map from R^{n} to itself and v = {v_{1}, …, v_{n}} is a linear basis of R^{n}, then the matrix of f with respect to the basis v is the n × n matrix M whose j-th column, for 1 ≤ j ≤ n, is the column of coordinates of f(v_{j}) relative to v, i.e.,

f(v_{j}) = ∑_{i=1}^{n} M_{ij}v_{i}, 1 ≤ j ≤ n .

Definition. If p = {p_{0}, …, p_{n}} is an affine basis of R^{n} and f is an affine map from R^{n} to itself then the affine matrix of f with respect to the affine basis p is the (n + 1) × (n + 1) matrix M whose j-th column, for 0 ≤ j ≤ n, is the column of barycentric coordinates of f(p_{j}) relative to p, i.e.,

f(p_{j}) = ∑_{i=0}^{n} M_{ij}p_{i} with ∑_{i=0}^{n} M_{ij} = 1 , 0 ≤ j ≤ n .

Proposition. If p is a point of R^{n} having barycentric coordinates (x_{0}, …, x_{n}) relative to the affine basis p and if f is an affine map having matrix M relative to p, then f(p) is the point of R^{n} having barycentric coordinates (y_{0}, …, y_{n}) relative to p where the vectors x and y, when regarded as columns, are related by the formula y = Mx.

Proof. Because f preserves barycentric combinations and p = x_{0}p_{0}+… x_{n}p_{n} with x_{0}+… +x_{n} = 1, it follows that

f(p) = ∑_{j} x_{j}f(p_{j}) = ∑_{j} x_{j} (∑_{i} M_{ij}p_{i})

= ∑_{ij} M_{ij}x_{j}p_{i} = ∑_{i} (∑_{j} M_{ij}x_{j}) p_{i}

= ∑_{i} y_{i}p_{i} where y = Mx

One needs to check that the last line is indeed a barycentric combination of the p_{i}, i.e., that y_{0} + … + y_{n} = 1. This follows from the fact that y is the x-barycentric combination of the (weight 1) columns of M

Exercises due Monday, March 15

1. Show that the map φ : R^{2} → R^{3} given by (x_{1}, x_{2}) ↦ (x_{1}, x_{2}, 1 − x_{1} − x_{2}) is an affine map.

2. Conclude from the first exercise that if τ is translation of R^{2} by the vector a = (a_{1}, a_{2}), then φ(τ(x)) = φ(x) + ã where ã is the weight 0 triple (a_{1}, a_{2}, a_{3}) with a_{3} = −a_{1} − a_{2}.

3. (Continuing) Find the affine matrix of the translation τ.

4. Find the affine matrix of the half turn of R^{2} about the point c, i.e., the affine transformation x ↦ 2c − x.

5. Show that if M is the affine matrix of the affine transformation f(x) = Ux + v of R^{2}, then det M = det U.