# Affine Transformations Affine transformations are very similar to linear transformations, only we remove the constraint that they must keep the origin fixed, meaning they allow for **translation**. #### Great intuition From this video [here](https://youtu.be/E3Phj6J287o?t=241) we can get a great intuition, namely that *affine transformations* are just a *subset* of all possible *linear operations* in $n+1$ dimensions. #### Affine combination Very similar to a [Linear Combination](Linear%20Combination.md), only now we have an additional **constraint** on the coefficients. So if we are looking at a linear combination of vectors $\vec{v}_1, \dots, \vec{v}_k \in \mathbb{R}^n$ it will take the form: $\lambda_1 \vec{v}_1 + \dots + \lambda_k \vec{v}_k$ Where $\lambda_1 , \dots, \lambda_k \in \mathbb{R}$. An affine combination imposes the constraint that the coefficients must sum to one: $\sum_{j=1}^k \lambda_j = 1$ Note that if we have the additional constraint that all coefficients must be positive then we have a **[convex-combination](convex-combination.md)**. #### Affine Hull (Affine Span) See [here](https://en.wikipedia.org/wiki/Affine_hull). Smallest space containing all affine combinations of vectors. See [this part](https://youtu.be/fWRm9dISpNk?t=437) of video.  --- Date: 20210527 Links to: [Linear Transformations](Linear%20Transformations.md) References: * [What are affine transformations? ](https://www.youtube.com/watch?v=E3Phj6J287o) * [wikipedia](https://en.wikipedia.org/wiki/Affine_transformation) * [Affine subspaces and transformations - 01 - affine combinations](https://www.youtube.com/watch?v=fWRm9dISpNk) * See Lecture notes [here](https://drive.google.com/file/d/1jTh5dZdVhoCzLkr_YRNLIurUjMaXkm3j/view?usp=sharing)