0

The Set $\{ Ax + b | Fx = g \}$, is it affine? How can I prove it?

My answer is yes, the intuition is that $\{ x | Fx = g \}$ is a solution space of equation $Fx = g$, thus it is a linear subspace.

The $Ax + b$ is a linear transformation plus a translation. So the final result is a affine set.

Welcome to comment on my understanding.

user2262504
  • 954

1 Answers1

0

Hint:
1. A set is an affine subspace, iff it is the kernel of an affine transformation $T: x \mapsto Mx+c$
2. An image of an affine subspace under an affine transformation is an affine subspace.

AlexR
  • 24,905
  • I don't understand the 1., what doest it mean "it is the kernel of an affine transformation"? I havenot met with "kernel" yet. Could you please explain in more detail? – user2262504 Sep 09 '14 at 01:05
  • Well, $\ker T := {x | Tx=0 }$. You might know it as a "null-space"? – AlexR Sep 09 '14 at 02:10
  • yes, I understand the concept of null-space, is "kernel" here same as null-space(solution space of $Tx = b $)? – user2262504 Sep 09 '14 at 03:39
  • @user2262504 Kernel is the solution space of $Tx = 0$. But an affine transformation with kernel ${Ax = b}$ is $Tx = Ax - b$. – AlexR Sep 10 '14 at 06:52