Table of Contents
A linear transformation is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces. In other words, it preserves linear combinations.
Additivity: f(x+y)=f(x)+f(y)
Homogeneity: f(ax)=af(x)
These claims are equivalent to:
f(a1x1+...+anxn) =a1f(x1)+...+anf(xn) (for any vectors x1,..,xn and scalars a1,..,an)
Consider V and W as 2-dimensional vector spaces. Every linear transformation from V to W can be represented as a matrix; this is useful because it allows concrete calculations. Conversely, matrices yield examples of linear transformations: if A is a real m-by-n matrix, then the rule f(x) = Ax describes a linear transformation Rn → Rm in an Euclidean space.
The composition of linear transformations is linear: if f : V → W and g : W → Z are linear, then so is g o f : V → Z.
If f1 : V → W and f2 : V → W are linear, then so is their sum f1 + f 2 [9].
If f : V → W is linear and a is an element of the ground field K, then the map af, defined by (af) (x) = a (f(x)), is also linear.
In the finite dimensional case (and if bases have been chosen), the composition of linear maps corresponds to the multiplication of matrices, the addition of linear maps corresponds to the addition of matrices, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.
The following sections will show you what utitly methods the Tensegrity Graph Framework provides to ease working with transformations.
© 2004, 2005 Tensegrity Software GmbH