Linear Transformations

Tags

math

A transformation is linear when both the following formulae hold.

\begin{align} T(k \begin{pmatrix}

x \\\ y

\end{pmatrix}) &=

k T
\begin{pmatrix}
  x \\\ y
\end{pmatrix} \\\\

T( \begin{pmatrix}

x\_1 \\\ y\_1

\end{pmatrix} + \begin{pmatrix}

x\_2 \\\ y\_2

\end{pmatrix}) &=

T(
\begin{pmatrix}
  x\_1 \\\ y\_1
\end{pmatrix})

+ T( \begin{pmatrix}

x\_2 \\\ y\_2

\end{pmatrix}) \end{align}

This can be asserted trivially be inserting possible values for \( (x,y) \) and checking whether these conditions hold.