Question

Derive the matrix of translation in 2-dimensional plane? Why its first 2 column is identity matrix

Answer 1

Answer on Question #42822 - Math - Linear Algebra

Problem. Derive the matrix of translation in 2-dimensional plane. Why its first 2 column is identity matrix?

Solution. A translation is a function that moves every point a constant distance in a specified direction.

We rewrite vector $\left[ \begin{array}{l}x\\ y \end{array} \right]$ in 2-dimensional using 3 coordinates as $\left[ \begin{array}{l}x\\ y\\ 1 \end{array} \right]$. Then the translation of vector $a = \left[ \begin{array}{l}a_1\\ a_2\\ 1 \end{array} \right]$ on vector $a = \left[ \begin{array}{l}b_1\\ b_2\\ 1 \end{array} \right]$ can be written using translation matrix $T_{a} = \left[ \begin{array}{lll}1 & 0 & a_{1}\\ 0 & 1 & a_{2}\\ 0 & 0 & 1 \end{array} \right]$ as

$$T _ {a} b = \left[ \begin{array}{c c c} 1 & 0 & a _ {1} \\ 0 & 1 & a _ {2} \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{c} b _ {1} \\ b _ {2} \\ 1 \end{array} \right] = \left[ \begin{array}{c} a _ {1} + b _ {1} \\ a _ {2} + b _ {2} \\ 1 \end{array} \right] = a + b.$$

First column in translation matrix correspond to rotation and scaling of vectors. A translation just moves vectors, so first 2 columns in the matrix of translation is identity matrix.

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