Question

Why is an affine transformation (f(x) = Wx.+ b) sometimes called a “linear transformation” (f(x) = Wx)? Hint: Consider their common properties.

Please explain the answer in detail and in good hand writing! Thanks a lot!

Answer 1

Both affine transformation $f(x)=Wx+b$ as well as linear transformation $f(x)=Wx$ describes mappings which preserve the linear structure of a space, meaning the way scaling the length of a vector parameterizes a line. If you apply a linear mapping, the image will still be a line.

Now a linear map is called a linear transformation if $T(\alpha x+\beta y)=\alpha T(x)+\beta T(y)$ for any scalars
a

Hence in the affine transformation $f(x)=Wx+b$ , if $b=0$

We get $f(x)=Wx$

And

$f( \alpha x+\beta y)=W( \alpha x+\beta y) \\~~~~~~ ~~~~~~~~~~~~~~=\alpha Wx+\beta Wy \\~~~~~~ ~~~~~~~~~~~~~~=\alpha f(x)+\beta f(y)$

Hence, the affine transformation $f(x)=Wx+b$ is called linear transformation $f(x)=Wx$ .