affine

definition

• of the form $A=\{x\in X:x=v+x^{(0)},v\in \mathcal{V}\}=x^{(0)}+\mathcal{V}$
  • $\mathcal{V}$ is subspace of X
  • obtained by adding a single vector to the span of a set of vectors
• Subspace are just affine space containing the origin, but not necessary have to pass through it

linear and affine functions

• a function f is affine IFF the function $f(x)=a^{T}x+b$ is linear, for some unique pair (a,b); $a\in R^n$ and $b\in R$
• function is linear IFF $b=0$
• affine = linear + constant
• difference btw a linear function & affine function:
  • a linear function always passes through the origin
  • affine function does not necessarily do so
• find a and b:
  • $a_i=f(e_i)-b$ with $e_i$ is the $i-th$ unit vector
  • $b=f(0)$

linear map

• $a\in R^n$ can be viewed as linear map from input space $R^n$ to output space $R$