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We have a matrix

$$\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$$

This matrix represents a transformation in homogeneous coordinates. My question is whether the above matrix is affine or not and an example too for this.

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It is not necessarily affine. An affine matrix in homogeneous coordinates has a form like:

$$\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ 0 & 0 & 1 \end{bmatrix}$$

(assuming you use a column vector convention). Here, the upper-left 2×2 submatrix is the linear part, and $(a_{13}, a_{23})$ is the translation vector of the affine transform.

If the lower row of the matrix has some values other than $[0, 0, 1]$, then it is in general a projective transform, not an affine one.

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