You have used matrices to calculate the results of certain rotations and reflections. Which ones? Are translations calculated using matrices?
I know of the different types of reflections, like across the x-axis, y-axis, origin, and y=x, but I haven’t heard of using matrices to calculate translations… does someone mind explaining to me? Thank you in advance!
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If you want to describe a translation in say $\mathbb{R}^2$ you may use a $3\times3$ matrix as follows:
For a translation by a vector $\vec{x}=(a,b)$ , $T_{\vec{x}}=\begin{pmatrix} 1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}$
So $T_{\vec{x}}\cdot (\alpha,\beta,1)^t=(a+\alpha,b+\beta,1)$ which is like $(\alpha,\beta)+\vec{x}=(a+\alpha,b+\beta)$
No, translations can't be computed via the same matrix-multiplication process.
The reason is that matrix multiplication will always map the vector $\vec0$ to itself, whereas a translation doesn't (except when the translation vector is $\vec0$ itself).
For all $a,b,c,d,\qquad$ $$\pmatrix{a & b \\c & d\\}\pmatrix{0 \\0 \\}=\pmatrix{0 \\0 \\}$$ But $$\pmatrix{0 \\0 \\}+\vec v=\vec v$$