properties of an affine transformation
is one-to-one iff is.
is onto iff is.
Suppose is onto. Let , so there are such that . Since is onto, there are with and . So . Hence is onto. Conversely, assume be onto, and pick . Take an arbitrary point and set . There is such that , since is onto. Let such that . Then . But is a bijection, we must have , showing that is onto. ∎
is a bijection iff is.
A bijective affine transformation is an affine isomorphism.
Suppose an affine transformation is a bijection. We want to show that is an affine transformation. Pick any , then
By the corollary above, is bijective, and hence a linear isomorphism. So
This shows that is an affine transformation whose assoicated linear transformation is . ∎
Two affine spaces associated with the same vector space are affinely isomorphic.
In fact, all we need to do is to show that is isomorphic to , where is given by . Pick any , then is a bijection. For any , there is a unique such that . Then , showing that is the associated linear transformation of . ∎
Any affine transformation is a linear transformation between the corresponding induced vector spaces. In other words, if is affine, then is linear.
Suppose are such that , or . Then
which is equivalent to .
Next, suppose , or , where . Then
which is equivalent to . ∎
|Title||properties of an affine transformation|
|Date of creation||2013-03-22 18:31:53|
|Last modified on||2013-03-22 18:31:53|
|Last modified by||CWoo (3771)|